A rectangular prism made of wood has a length of 10 centimeters, a width of 8 centimeters, and a height of 12 centimeters. A rectangular hole with a length of 2 centimeters and a width of 3 centimeters is cut through the prism as shown. What is the volume of the resulting figure? A. 882 cubic centimeters B. 888 cubic centimeters C. 960 cubic centimeters D. 1,032 cubic centimeters

Answer:the cost of one child's ticket is $6

the cost of one adult ticket is $9

Step-by-step explanation:

Let x represent the cost of one child's ticket

Let y represent the cost of one adult ticket.

Tickets for a minor league baseball game for an adult and two children cost a total of $21. This means that

2x + y = 21 - - - - - - - - - - 1

The adult ticket is three dollars more than a child's ticket. This means that

y = x + 3

Substituting y = x + 3 into equation 1, it becomes

2x + x + 3 = 21

3x + 3 = 21

3x = 21 - 3 = 18

x = 18/3 = 6

y = x + 3 = 6 + 3

y = 9

Answer:

a) [tex] P(A|B) = \frac{15/83}{44/83} =\frac{15}{44}=0.341[/tex]

b) [tex] P(B|A) = \frac{29/83}{44/83} =\frac{29}{44}=0.659[/tex]

c)  A. A student given a $1 bill is more likely to have kept the money.

Because the probability 0.659 is atmoslt two times greater than 0.341

Step-by-step explanation:

Assuming the following table:

                                                     Purchased Gum      Kept the Money   Total

Students Given 4 Quarters              25                              14                      39

Students Given $1 Bill                       15                               29                    44

Total                                                   40                              43                     83

a. find the probability of randomly selecting a student who spent the money, given that the student was given a $1 bill.

For this case let's define the following events

B= "student was given $1 Bill"

A="The student spent the money"

For this case we want this conditional probability:

[tex] P(A|B) =\frac{P(A and B)}{P(B)}[/tex]

We have that [tex] P(A)= \frac{40}{83} , P(B)= \frac{44}{83}, P(A and B)= \frac{15}{83}[/tex]

And if we replace we got:

[tex] P(A|B) = \frac{15/83}{44/83} =\frac{15}{44}=0.341[/tex]

b. find the probability of randomly selecting a student who kept the money, given that the student was given a $1 bill.

For this case let's define the following events

B= "student was given $1 Bill"

A="The student kept the money"

For this case we want this conditional probability:

[tex] P(A|B) =\frac{P(A and B)}{P(B)}[/tex]

We have that [tex] P(A)= \frac{43}{83} , P(B)= \frac{44}{83}, P(A and B)= \frac{29}{83}[/tex]

And if we replace we got:

[tex] P(B|A) = \frac{29/83}{44/83} =\frac{29}{44}=0.659[/tex]

c. what do the preceding results suggest?

For this case the best solution is:

A. A student given a $1 bill is more likely to have kept the money.

Because the probability 0.659 is atmoslt two times greater than 0.341

Using the principles of conditional probability, we find that a student given a $1 bill is more likely to spend the money (probability approximately 0.641) than keep it (probability approximately 0.359). Therefore, option B is the correct interpretation of these results.

To answer this question, we should first understand that this problem is fundamentally about conditional probability, the probability of an event given that another event has occurred. Let's take this step by step.

Part a: Here, we want to find the probability of selecting a student who spent the money, given that the student was given a $1 bill. This number would be the number of $1 bill students who bought gum divided by the total number of $1 bill students. This equates to 25/(25+14) = 0.641. So, the probability is approximately 0.641.

Part b: In this situation, we're looking for the probability of selecting a student who kept the money, given that the student was given a $1 bill. This would be the number of $1 bill students who kept the money divided by the total number of $1 bill students, or 14/(25+14) = 0.359. The probability is approximately 0.359.

Part c: These results suggest that the appropriate solution is B: 'A student given a $1 bill is more likely to have spent the money than a student given four quarters.'

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Answer:

[tex]H(t)=-20\text{cos}(\frac{\pi}{12}t)+60[/tex]

Step-by-step explanation:

We have been given that the desert temperature, H, oscillates daily between 40◦F at 5 am and 80◦F at 5 pm. We are asked to write a formula H in terms of t, measured in hours from 5 am.

We will use cosine function to write our required formula.

[tex]y=A\text{cos}[B(x-C)]+D[/tex], where,

A = Amplitude,

[tex]\text{Period}=\frac{2\pi}{|B|}[/tex]

C = Phase shift,

D = Vertical shift.

First of all, we will find amplitude using maximum and minimum values as:

[tex]A=\frac{\text{Maximum value}-\text{Minimum value}}{2}[/tex]

[tex]A=\frac{80-40}{2}[/tex]

[tex]A=\frac{40}{2}[/tex]

[tex]A=20[/tex]

Since period is 24 hours (5 am to 5 pm), so let us find B as:

[tex]24=\frac{2\pi}{|B|}[/tex]

[tex]B=\frac{2\pi}{24}[/tex]

[tex]B=\frac{\pi}{12}[/tex]

[tex]\text{Vertical shift}=\frac{\text{Maximum value}+\text{Minimum value}}{2}[/tex]

[tex]D=\frac{80+40}{2}=\frac{120}{2}=60[/tex]

There is no phase shift.

Since temperature is minimum when [tex]t=0[/tex], so we will use negative cosine as:

[tex]H(t)=-20\text{cos}(\frac{\pi}{12}t)+60[/tex]

Therefore, our required function would be [tex]H(t)=-20\text{cos}(\frac{\pi}{12}t)+60[/tex].

To model the desert temperature, use the sinusoidal function H(t) = 20 * cos(π/12 * t) + 60, which captures daily temperature changes between 40°F and 80°F, with parameters tuned to fit the given data.

To write a formula for the desert temperature H in terms of t, measured in hours from 5 am, we can use a sinusoidal function that models the daily temperature oscillation.

We know:

The general form for the sinusoidal function is:

H(t) = A * sin(B*(t - C)) + D

Where:

From the given data:

Thus, the sinusoidal function becomes:

H(t) = 20 * cos(π/12 * t) + 60.

Answer:

The dimensions of poster are 32.5 in wide and 40.5 in tall.

Step-by-step explanation:

Given:

A poster is 8 in taller than it is wide.

It is mounted on a backing board that provides a 2 in border on each side of the poster.

The area of the backing board is 308 in².

Now, to find the dimensions of the poster.

Let [tex]x[/tex] be the length of the poster.

And [tex]y[/tex] be the width of the poster.

As given, poster is 8 in taller than it is wide.

So,

[tex]x=y+8[/tex]   ......(1)

Area = 308 in².

So, it is mounted on a backing board that provides a 2 in border on each side of the poster.

According to question:

[tex]2\times (2(y+4))+2\times (2\times x)=308[/tex]

Now. substituting the value from equation (1) in the place of [tex]x[/tex] we get:

[tex]2\times (2y+8)+2(2\times (y+8))=308[/tex]

[tex]2\times (2y+8)+2(2y+16)=308[/tex]

[tex]4y+16+4y+32=308[/tex]

[tex]8y+48=308[/tex]

Subtracting both sides by 48 we get:

[tex]8y=260[/tex]

Dividing both sides by 8 we get:

[tex]y=32.5\ in.[/tex]

The width of the poster = 32.5 in.

Now, substituting the value of [tex]y[/tex] in equation (1):

[tex]x=y+8[/tex]

[tex]x=32.5+8[/tex]

[tex]x=40.5\ in.[/tex]

Length of the poster = 40.5 in.

Therefore, the dimensions of poster are 32.5 in wide and 40.5 in tall.

the maximum number of boxes that will fit on the shelf is 21.

To find out the maximum number of boxes that will fit on the shelf, we need to calculate the area of the shelf and the area of each box, then divide the total area of the shelf by the area of each box.

Given:

- Shelf dimensions: 65 cm x 35 cm

- Box dimensions: 180 mm x 60 mm

First, let's convert all dimensions to the same unit. Let's choose millimeters for consistency:

Shelf dimensions:

- Length: [tex]\(65 \, \text{cm} \times 10 \, \text{mm/cm} = 650 \, \text{mm}\)[/tex]

- Width: [tex]\(35 \, \text{cm} \times 10 \, \text{mm/cm} = 350 \, \text{mm}\)[/tex]

Now, let's calculate the area of the shelf:

[tex]\[ \text{Shelf area} = \text{Shelf length} \times \text{Shelf width} \][/tex]

[tex]\[ \text{Shelf area} = 650 \, \text{mm} \times 350 \, \text{mm} \][/tex]

[tex]\[ \text{Shelf area} = 227,500 \, \text{mm}^2 \][/tex]

Now, let's calculate the area of each box:

[tex]\[ \text{Box area} = \text{Box length} \times \text{Box width} \][/tex]

[tex]\[ \text{Box area} = 180 \, \text{mm} \times 60 \, \text{mm} \][/tex]

[tex]\[ \text{Box area} = 10,800 \, \text{mm}^2 \][/tex]

Now, to find out the maximum number of boxes that will fit on the shelf, we divide the shelf area by the area of each box:

[tex]\[ \text{Maximum number of boxes} = \frac{\text{Shelf area}}{\text{Box area}} \][/tex]

[tex]\[ \text{Maximum number of boxes} = \frac{227,500 \, \text{mm}^2}{10,800 \, \text{mm}^2} \][/tex]

[tex]\[ \text{Maximum number of boxes} \approx 21.06 \][/tex]

Since we can't have a fraction of a box, the maximum number of boxes that will fit on the shelf is 21.

complete question given below:

The campsite shop sells boxes of Funshine Cereal.

The base of each box is a 180 mm x 60 mm rectangle.

The shelf where the boxes are displayed is a 65 cm x 35 cm rectangle.

Work out the maximum number of boxes that will fit on the shelf.

the maximum number of boxes that will fit on the shelf is 21.

To find the maximum number of boxes that fit on the shelf, we divide the shelf area by the area of each box.

Given:

- Shelf dimensions: 65 cm x 35 cm

- Box dimensions: 180 mm x 60 mm

First, let's convert the shelf dimensions to millimeters for consistency:

Shelf dimensions:

- Length: [tex]\(65 \, \text{cm} \times 10 \, \text{mm/cm} = 650 \, \text{mm}\)[/tex]

- Width: [tex]\(35 \, \text{cm} \times 10 \, \text{mm/cm} = 350 \, \text{mm}\)[/tex]

Now, calculate the area of the shelf:

[tex]\[ \text{Shelf area} = \text{Shelf length} \times \text{Shelf width} \]\[ \text{Shelf area} = 650 \, \text{mm} \times 350 \, \text{mm} \]\[ \text{Shelf area} = 227,500 \, \text{mm}^2 \][/tex]

Next, calculate the area of each box:

[tex]\[ \text{Box area} = \text{Box length} \times \text{Box width} \]\[ \text{Box area} = 180 \, \text{mm} \times 60 \, \text{mm} \]\[ \text{Box area} = 10,800 \, \text{mm}^2 \][/tex]

Now, find the maximum number of boxes that fit on the shelf:

[tex]\[ \text{Maximum number of boxes} = \frac{\text{Shelf area}}{\text{Box area}} \]\[ \text{Maximum number of boxes} = \frac{227,500 \, \text{mm}^2}{10,800 \, \text{mm}^2} \]\[ \text{Maximum number of boxes} \approx 21.06 \][/tex]

Since we can't have a fraction of a box, the maximum number of boxes that will fit on the shelf is 21.

The probable question maybe:

The campsite shop sells boxes of Funshine Cereal.

The base of each box is a 180 mm x 60 mm rectangle.

The shelf where the boxes are displayed is a 65 cm x 35 cm rectangle.

Work out the maximum number of boxes that will fit on the shelf.

Answer:

C.

is the right answer

Step-by-step explanation:

The value of the 4 in 64.723 is 100 times the value of the 4 in 9.048. The correct option is B

A system of writing numbers is known as a number system. It is the mathematical notation for consistently using digits or other symbols to represent the numbers in a given set.

It represents the arithmetic and algebraic structure of the numbers and gives each number a distinct representation.

Given that there are two numbers 64.723 and 9.048. The value of the number 4 in one number is 100 times the other. As it is at one-hundredth place of another number

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Answer:

The 3th answer is correct.

for x<2, form graph we know that if x=0 than y=-3.

For x<2 in 3th we have y=1/2x-3, so for x=0, y=(1/2)*0-3=0-3=-3. It fits.

For x=2, from graph we have that y=-2.

For x>=2 in 3th we have y=3x-8, so for x=2, y=3*2-8=-2. It fits.

These answer fit also i for first, but breakpoint for y from graph is point 2 not point -2, so the answer is 3th.

Answer:

Step-by-step explanation:

If angle 1 and angle 2 are supplementary, then

∠1 + ∠2 = 180°

If angle 2 and angle 3 are complementary, then

∠2 + ∠3 = 90°

We are given the values for angles 1 and 2, so:

(8x + 3) + ( 5x - 18) = 180 so

13x - 15 = 180 and

13x = 195 so

x = 15

Sub 15 in for x in angle 2 to find the measure of angle 2:

∠2 = 5(15) - 18 so

∠2 = 57°

That means that to find angle 3,

57° + ∠3 = 90 so

∠3 = 33°

The measure of angle 3 is found to be 33 degrees, using the relationships that angle 1 is supplementary to angle 2 and angle 2 is complementary to angle 3, along with their given expressions in terms of x.

To find the measure of angle 3, we must use the relationships between the angles given: angle 1 is supplementary to angle 2 and angle 2 is complementary to angle 3.

Since angle 1 and angle 2 are supplementary, their sum is 180 degrees. The equation is therefore:

(8x + 3) + (5x - 18) = 180

Solving for x gives:

13x - 15 = 180

x = 195 / 13

x = 15

Substitute x back into the expression for angle 2:

(5x - 18) = (5(15) - 18) = 57 degrees

Since angle 2 and angle 3 are complementary, they add up to 90 degrees:

angle 2 + angle 3 = 90

57 + angle 3 = 90

Subtract 57 from both sides to find angle 3:

angle 3 = 90 - 57

angle 3 = 33 degrees

Therefore, the measure of angle 3 is 33 degrees.

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Answer:

1 2/3

Step-by-step explanation:

Luis mixed all the trails thus:

4 2/3 + 1 1/4 + 1 1/3 + 3/4

= 15/3 + 5/4 + 4/3 + 3/4

= 60+15+16+9/12

Total trail mixed by Luis equals 100/12

He then divided 100/12 among his 5 friends thus:

100/12 ÷ 5

= 20/12

= 1 2/3 per friend.

Answer:

1  3/5

Step-by-step explanation:

the answer is 1.6 but you have to convert is to a fraction  because there are no  decimal numbers theres. if you convert it , it will be 1 3/5

Answer:

64%

Step-by-step explanation:

pain

The answer is B) 64%.

Probability is a number that expresses the likelihood or chance that a specific event will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.

Let X be the number of students out of 40 who like to hike. We know that P(X ≤ 15) = 0.36 and we want to find P(X ≥ 16).

Note that P(X ≥ 16) is the complement of P(X ≤ 15). That is,

P(X ≥ 16) = 1 - P(X ≤ 15) = 1 - 0.36 = 0.64

Therefore, the answer is B) 64%.

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Answer:

(5/3, 41/3)

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Equality Properties

Algebra I

Step-by-step explanation:

Step 1: Define Systems

y = 10x - 3

y = 7x + 2

Step 2: Solve for x

Substitution

Step 3: Solve for y

Step 4: Check

Verify the solution set to the systems of equations by graphing the systems.

Where the 2 lines intersect is the solution set to the systems of equations.

We see graphically that point (1.666667, 13.66667), equivalent to (5/3, 41/3).

∴ (5/3, 41/3) or x = 5/3 and y = 41/3 is the solution to the systems of equations.

Final answer:

The solution to the system of equations y = 10x - 3 and y = 7x + 2 is x = 5/3 and y = 19/3.

Explanation:

The solution of the system of equations given by y = 10x - 3 and y = 7x + 2 can be found by setting the two expressions for y equal to each other since they both equal y. This yields the equation 10x - 3 = 7x + 2. Solving for x, we first subtract 7x from both sides to get 3x - 3 = 2. Then we add 3 to both sides to get 3x = 5. Finally, dividing both sides by 3 gives us x = 5/3. Substituting x back into either original equation, we get y = 10(5/3) - 3 or y = 7(5/3) + 2, both of which simplify to y = 19/3. Therefore, the solution to the system is x = 5/3 and y = 19/3.

Answer:

The distance that the conveyor belt moves the bottle is less than 90 meters

Step-by-step explanation:

Speed of the conveyor belt

120 cm/s

Time it takes to move the conveyor belt less than 1.2 minutes to move the bottle from the loading dock to the unloading dock.

But 60s = 1 minute

This gives

1.2 minutes = 1.2×60s seconds = 72 seconds

But speed = distance/time

so that distance = speed × time = 120cm/s ×(<72s)

From where distance = <8640cm<90m (90m×100=9000cm)

Time it takes to move the conveyor belt more than 1.1 minutes to move the bottle from the loading dock to the unloading dock.

Again 60s = 1 minute

This gives

1.1 minutes = 1.1×60s seconds = 66 seconds

Time it takes is more than 66s

But speed = distance/time

so that distance = speed × time = 120cm/s × (>66s)

From where distance >7920cm

However

7920cm < 90m (90m×100=9000cm)

So that

The distance that the conveyor belt moves the bottle is less than 90 meters

Answer:

The cubic polynomial is: x³ - x² - 6x.

Step-by-step explanation:

Given the degree and the roots of the polynomial we can find it.

An n - degree polynomial has n roots.

Here, given that the degree of the polynomial is 3 and three roots are given. Also, if (x - a) is a factor of a polynomial then x = a is a root of the polynomial. The converse is also true.

Since, the roots of the polynomial are given to -2, 0, 3 then it should have had the following factors.

(x + 2)(x - 0)(x - 3) = 0

Multiplying them we get:

⇒ [tex]$ (x^2 + 2x)(x - 3) $[/tex]

[tex]$ = x^3 - 3x^2 + 2x^2 - 6x $[/tex]

[tex]$ = x^3 - x^2 - 6x $[/tex] which is the required cubic polynomial.

Hence, the answer.

Answer:

B. Stratified sampling, because there are specific subgroups to investigate.

Step-by-step explanation:

Stratified Sampling is a sampling method and is used when the population has subgroups with different characteristics. Random sampling is applied in each of the sub-groups proportional to their size and then these samples combined together to create sample of the study.

Since there are three towns with people from different occupations ( retirees,business owners, office workers), it is better to use stratified sampling method.

The correct answer is option B). Stratified sampling, because there are specific subgroups to investigate.

To determine the most appropriate sampling method, one must consider the composition of the population and the goal of the study. In this scenario, the population consists of three distinct groups: retirees in town A, business owners in town B, and office workers in town C. Each group represents a different stratum within the overall population of the county.

Stratified sampling is a method that divides the population into subgroups, or strata, based on certain characteristics, such as age, occupation, or town of residence in this case. This method ensures that each subgroup is represented in the sample in proportion to its size in the overall population. Here's why stratified sampling is the most appropriate method for this situation:

1. Representation: Stratified sampling ensures that each town's unique demographic is represented in the sample. This is important because each group may have different voting patterns.

2. Precision: By sampling each stratum separately, the estimates of the average number of votes for each candidate can be more precise because the variance within each subgroup is likely to be lower than the variance across the entire population.

3. Comparability: Stratified sampling allows for direct comparison between the different strata, which in this case are the three towns.

4. Efficiency: If the strata are chosen such that the variance within each stratum is minimized, the overall variance of the estimate can be reduced, leading to a more efficient estimate.

Now, let's consider the other options:

A. Simple random sampling: While this method is unbiased, it may not provide representation from each of the three towns in proportion to their population sizes. This could lead to an inaccurate estimate if one group's voting behavior is significantly different from the others.

C. Systematic sampling: This method involves selecting elements from the population at regular intervals. It can be difficult to implement if there is no clear ordering of the population, and it may not ensure representation from each town.

D. Cluster sampling: This method would be appropriate if the population were spread across a wide area and could be divided into clusters that are more convenient to sample. However, since the population is already divided into distinct groups within specific towns, stratified sampling is a better choice.

E. All the above sampling methods: While each method has its own merits, not all are equally suitable for this particular scenario. Stratified sampling is the most appropriate because it specifically addresses the heterogeneity of the population across the three towns.

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m = 1/7 is the slope

(x,y) = (0,0) is the origin the line goes through

y = mx+b

0 = (1/7)*0 + b

0 = 0+b

b = 0 is the y intercept

y = mx+b

y = (1/7)x+0

y = (1/7)x is the equation of the line

----------------------

To plot the equation of this line, mark the point (0,0) first.

Then move up 1 unit and to the right 7 units to arrive at (7,1) as the second point.

Draw a straight line through (0,0) and (7,1) as shown in the diagram below.

Point P is (0,0) and point Q is (7,1)

Points A through E in the same diagram represent the answer choices A through E.

Of the answer choices, only point D is on this line, so point D is the answer.

---------------

A non-visual way to find the answer is to plug each (x,y) coordinate from each answer choice into the equation we found above.

So for choice A we plug in x = 0 and y = 7

y = (1/7)*x

7 = (1/7)*0

7 = 0

we end up with a false equation, so choice A is ruled out. Similar stories happen with B, C, and E as well.

With choice D however, we plug in x = 14 and y = 2, and we get...

y = (1/7)*x

2 = (1/7)*14

2 = 14/7

2 = 2

Since we get a true equation, this confirms that (14,2) is on the graph of y = (1/7)x.

Defining the wrong choice:

Therefore, "Choice D" is the correct choice.

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Step-by-step explanation:

For a : Speed = distance / time

Distance covered = y2 - y1 = 20 - 15 = 5

Time taken = x2 - x1 = 8 - 0 = 8

Speed in part a : 5/8 = 0.625

4. It takes her 32 minutes to get home. We can see from the graph that it is the total time taken throughout the whole journey.

Answer:the fraction of Mary's garden that has tulips is 2/3

Step-by-step explanation:

Let x represent the full area if Mary's garden.

Mary plants roses in 1/4 of her garden. This means that the portion of the garden on which she planted roses would be 1/4 × x = x/4

She also plants some tulips in her garden. She has 1/12 of the garden left to plant more flowers. This means that the portion if the garden that she has left to plant more flowers would be 1/12 × x = x/12

The portion of the garden on which she planted tulips would be

x - (x/4 + x/12) = x - (3x + x)/12 = x - 4x/12

x - x/3 = (3x - x)/3 = 2x/3

Therefore, the fraction of Mary's garden that has tulips would be

(2x/3) /x = 2/3

Final answer:

Mary has 2/3 of her garden planted with tulips.

Explanation:

To find the fraction of Mary's garden that has tulips, we need to first determine the fraction of her garden that she has planted with roses. We are given that Mary plants roses in 1/4 of her garden, so the fraction of her garden that she has planted with roses is 1/4. Since she has 1/12 of her garden left to plant more flowers, the fraction of her garden that she has planted with tulips is the remaining fraction, which is 1 - 1/4 - 1/12.

Let's simplify this expression. First, find a common denominator for 1/4 and 1/12, which is 12. Rewrite 1/4 with the denominator 12 as 3/12, and rewrite 1/12 with the denominator 12 as 1/12. Now we can subtract these fractions:

1 - 3/12 - 1/12 = 12/12 - 3/12 - 1/12 = 8/12 = 2/3

Therefore, 2/3 of Mary's garden has tulips.

Christopher Columbus landed on an island in the Bahamas, which he named San Salvador, on October 12, 1492, mistakenly believing he had reached Asia. This event is considered a pivotal moment in history, marking the beginning of European exploration and colonization of the Americas.

Christopher Columbus, an Italian explorer sponsored by Spain, embarked on his voyage to find a direct sea route to Asia by sailing west. Contrary to his expectations, he landed in the Americas on October 12, 1492. Columbus made landfall on an island in the Bahamas, which the native Lucayans called Guanahani. He renamed it San Salvador. Following this, Columbus explored other islands in the Caribbean, including an island he named Hispaniola (present-day Dominican Republic and Haiti), still believing he had reached the East Indies.

Columbus’ mistaken belief that he had landed in Asia led to the indigenous peoples he encountered being called “Indios”, which is the origin of the term “Indian” for native peoples of the Americas. Despite his error, Columbus's voyages are considered a pivotal moment in history, marking the beginning of widespread European exploration and colonization of the Americas.

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Answer:

B. and C.

General Formulas and Concepts:

Calculus

Differentiation

Integration

U-Substitution

Step-by-step explanation:

*Note:

It seems like B and C are both the same answer.

Let's define our answer choices:

a.  [tex]\displaystyle \int {\sqrt{x - 1}} \, dx[/tex]

b.  [tex]\displaystyle \int {\frac{1}{\sqrt{1 - x^2}}} \, dx[/tex]

c.  [tex]\displaystyle \int {\frac{1}{\sqrt{1 - x^2}}} \, dx[/tex]

d.  [tex]\displaystyle \int {x\sqrt{x^2 - 1}} \, dx[/tex]

Let's run u-substitution through each of the answer choices:

a.  [tex]\displaystyle u = x - 1 \rightarrow du = dx \ \checkmark[/tex]

∴ answer choice A can be evaluated with a simple substitution.

b.  [tex]\displaystyle u = 1 - x^2 \rightarrow du = -2x \ dx[/tex]

We can see that this integral cannot be evaluated with a simple substitution. In fact, this is a setup for an arctrig integral.

∴ answer choice B cannot be evaluated using a simple substitution.

C.  [tex]\displaystyle u = 1 - x^2 \rightarrow du = -2x \ dx[/tex]

We can see that this integral cannot be evaluated with a simple substitution. In fact, this is a setup for an arctrig integral.

∴ answer choice C cannot be evaluated using a simple substitution.

D.  [tex]\displaystyle u = x^2 - 1 \rightarrow du = 2x \ dx \ \checkmark[/tex]

Using a little rewriting and integration properties, this integral can be evaluated using a simple substitution.

∴ answer choice D can be evaluated using a simple substitution.

Out of all the choices, we see that B and C cannot be evaluated using a simple substitution.

∴ our answer choices should be B and C.

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

Book: College Calculus 10e

Answer:

1, 2, and ±i√8

Step-by-step explanation:

0 = x⁴ − 3x³ + 10x² − 24x + 16

Using grouping:

0 = x⁴ − 3x³ + 2x² + 8x² − 24x + 16

0 = x² (x² − 3x + 2) + 8 (x² − 3x + 2)

0 = (x² + 8) (x² − 3x + 2)

0 = (x² + 8) (x − 1) (x − 2)

The roots are 1, 2, and ±i√8.

Answer: Christina's age is 13, Joanne's age is 42 and Devin's age is 21.

Step-by-step explanation: If Devin's age is represented by d, then Joanne's age will be 2d because Joanne is two times Devin's age. Also, if Devin is 8 years older than Christina, then Christina's age would be d - 8. If the sum of their ages is equal to 76, then;

2d + d + (d - 8) = 76

2d + d + d - 8 = 76

4d - 8 = 76

Add 8 to both sides of the equation

4d = 84

Divide both sides of the equation by 4

d = 21

Therefore, Devin is 21 years old

Joanne is 42 years old and

Christina is 13 years old.

Christina is 13 years old, Devin is 21 years old, and Joanne is 42 years old. This is found by defining Christina's age as x, then setting up and solving an equation based on the problem's provided relationships.

In this math problem, we have three individuals: Joanne, Devin, and Christina, and we know their ages in relation to each other's.

We know that Joanne's age is two times Devin's. We also know Devin is eight years older than Christina. Finally, we know the sum of all three ages is 76.

Let's denote Christina's age as x. Consequently, Devin's age would be x + 8 because he is 8 years older than Christina. As Joanne's age is two times Devin's age, Joanne's age would be 2*(x + 8).

The sum of their ages is 76, which gives us the following equation:
x (Christina's age) + x + 8 (Devin's age) + 2*(x + 8) (Joanne's age) = 76.

Solving this equation, we get:
4x + 24 = 76
Subtract 24 from both sides gives us 4x = 52, and dividing both sides by 4 gives x = 13.

So, Christina's age is 13, Devin's age is 13 + 8 = 21, and Joanne's age is 2*21 = 42.

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Step-by-step explanation:

Let the number be n.

When ​one-fourth of a number is added to ​one-fifth of the​ number, the sum is 18

          [tex]\frac{n}{4}+\frac{n}{5}=18\\\\\frac{5n+4n}{4\times 5}=18\\\\9n=4\times 5\times 18\\\\n=20\times 2\\\\n=40[/tex]

The number is 40.

Answer:

unknown number is 40

Step-by-step explanation:

 Let x be the unknown number

one-fourth of a number is added to ​one-fifth of the​ number

[tex]\frac{1}{4}x+\frac{1}{5} x[/tex]

sum is 18

[tex]\frac{1}{4}x+\frac{1}{5} x=18[/tex]

To solve for x take LCD 20

[tex]\frac{5}{20}x+\frac{4}{20} x=18\\ \frac{9}{20} x=18\\[/tex]

multiply both sides by 20

[tex]9x= 360[/tex]

divide both sides by 9

x=40

Answer:

[tex]2\sqrt {10}[/tex]

Step-by-step explanation:

Given:

[tex]x=3\sqrt 5[/tex]

[tex]y=\sqrt 5[/tex]

As seen from the triangle, the triangle is a right angled triangle. Two sides of the triangle are given and we are asked to find the third side.

'x' is the hypotenuse as this side is opposite side to the right angled.

'y' and 'z' are the two legs of the triangle.

Now, using pythagoras theorem,

[tex]x^2=y^2+z^2\\\\(3\sqrt 5)^2=(\sqrt 5)^2+z^2\\\\9\times 5=5+z^2\\\\45-5=z^2\\\\40=z^2\\\\z=\sqrt {40}\\\\z=2\sqrt {10}[/tex]

Therefore, the measure of the side 'z' is [tex]2\sqrt {10}[/tex].

Hence, the third option is correct.

Answer:

There are 7 dimes, 3 nickels, and 4 quarters.

Answer:

Mrs. Jackson worked 32 hours.

Step-by-step explanation:

Given:

Mrs.Jackson earned a $500 bonus for signing a 1 year contract to work as a nurse.

Her salary is $22 per hour.

If her first weeks check including the bonus is $1204.

Now, to find the hours Mrs. Jackson work.

Amount of bonus = $500.

Total salary = $1204.

So, we deduct the amount of bonus from the total salary:

[tex]1204-500=\$704.[/tex]

The remaining salary = $704.

As, given her salary is $22 per hour.

Now, to get the hours she work we divide the remaining salary $704 by $22:

[tex]704\div 22[/tex]

[tex]=32\ hours.[/tex]

Therefore, Mrs. Jackson worked 32 hours.

Answer:

Available information is not sufficient to solve the questions

Step-by-step explanation:

Let Cm = Computers Ordered by Company M

Let Cn = Computers Ordered by Company N

Let Pm = Printers Ordered by Company M

Let Pn = Printers Ordered by Company N

Cm + Pm = 50 --- Equation 1

Cn + Pn = 60 ----- Equation 2

1. Company M and N ordered the same number of computers

Here,

Cm = Cn

Substitute Cm for Cn in equation 2

Cm + Pm = 50

Cm + Pn = 60

Subtract Equation 2 from 1

Pm - Pn = 50 - 60

Pm - Pn = -10

Pm = Pn - 10

Substitute Pn-10 for Pm in equation 1

Cm + Pn - 10 = 50

Make Pn the subject of formula

Cm = 50+10-Pn

Cm = 60 - Pn

Hence, the number of computers ordered by Company M is 60 minus the number of printers ordered by Company n

2. Company N ordered 10 more printers than Company M.

Here,

Pn = 10 + Pm

Make Pm the subject of formula

Pm = Pn - 10

Substitute Pn-10 for Pm in equation 1

Cm + Pn - 10 = 50

Make Cm the Subject of formula

Cm = 50 + 10 - Pn

Cm = 60 - Pn

Hence, the number of computers ordered by Company M is 60 minus the number of printers ordered by Company n

Condition 1 and 2 points to the same result

Final answer:

Greg bought 12 books online. To find the number of books, set up the equation 6n + 3 = 75, subtract the shipping fee, and divide by the price per book.

Explanation:

Greg ordered books online for $6 each and paid a total of $75, which includes a $3 shipping fee. To determine the number of books Greg bought, you need to set up an equation that represents the total cost.

The cost of the books alone can be represented by 6 times the number of books (6n), and when you add the shipping cost of $3, it equals the total cost of $75. So the equation would be:

6n + 3 = 75

To solve for n, which is the number of books Greg purchased, follow these steps:

Subtract the shipping fee from the total cost: 75 - 3 = 72.

Divide the result by the cost per book: 72 ÷ 6 = 12.

Therefore, Greg bought 12 books.

Answer:

Lia needs [tex]7\frac{2}{4}\ cups[/tex] of flour and corn meal.

Step-by-step explanation:

Given:

Amount of flour needed = [tex]2\frac{3}{4} \ cups[/tex]

[tex]2\frac{3}{4} \ cups[/tex] can be Rewritten as [tex]\frac{11}{4}\ cups[/tex]

Amount of flour needed =  [tex]\frac{11}{4}\ cups[/tex]

Amount of Corn meal needed = [tex]4\frac{3}{4}\ cups[/tex]

[tex]4\frac{3}{4}\ cups[/tex] can be Rewritten as [tex]\frac{19}{4}\ cups[/tex]

Amount of Corn meal needed =  [tex]\frac{19}{4}\ cups[/tex]

We need to find the total amount of of flour and cornmeal she needs.

Solution:

Now we can say that;

the total amount of of flour and cornmeal she needs is equal to sum of Amount of flour needed and Amount of Corn meal needed.

framing in equation form we get;

the total amount of of flour and cornmeal she needs = [tex]\frac{11}{4}+\frac{19}{4}=\frac{11+19}{4}=\frac{30}{4}\ cups\ \ OR\ \ 7\frac{2}{4}\ cups[/tex]

Since Answers are not given:

Kindly chose the answer which contains below data.

Hence Lia needs [tex]7\frac{2}{4}\ cups[/tex] of flour and corn meal.

Answer:

Step-by-step explanation:

u = 3.4

stdev = 0.35

n = 25

E = u = 3.4

SD = [tex]\frac{stdev}{\sqrt{n} } =\frac{0.35}{\sqrt{25} }[/tex] = 0.07

The calculation of the 68% population covers with 1 standard deviation is as follows:

u - SD = 3.4 - 0.07 = 3.33

u + SD = 3.4 + 0.07 = 3.47

Range = (3.33, 3.47)

The calculation of the 95% population covers within 2 standard deviations is as follows:

u - 2SD = 3.4 - 2(0.07) = 3.26

u + 2SD = 3.4 + 2(0.07) = 3.54

Range = (3.26, 3.54)

The calculation of the 99.7% population covers within 3 standard deviations is as follows:

u - 3SD = 3.4 - 3(0.07) = 3.19

u + 3SD = 3.4 + 3(0.07) = 3.61

Range = (3.19, 3.61)

From the information, observe that the shape of the distribution is symmetrical.

Therefore, the graph is as shows the attached image.

This shows that approximately:

68% of the observations will have mean between 3.33 and 3.47

95% of the observations will have mean between 3.26 and 3.54

99.7% of the observations will have mean between 3.19 and 3.61