A bookcase has a mass of 30 kilograms. A book in the bookcase has a mass of 400 grams. How many books of the same mass would it take to equal the mass of the bookcase? A. 75 books B. 120 books C. 750 books D. 1,200 books

The expression that represents the statement is (56 - 14) ÷ 8.

To represent the given statement, "Subtract 14 from 56 and divide the result by 8," we need to follow these steps:

Step 1: Subtract 14 from 56: 56 - 14 = 42

Step 2: Divide the result by 8: 42 ÷ 8 = 5.25

So, the expression that represents the statement is (56 - 14) ÷ 8.

The correct answer is: (56 - 14) ÷ 8.

The complete question is here:

Which expression represents the statement shown? Subtract 14 from 56 and divide the result by 8. (56/ 8)-14 (14-56)/ 8 14-(56/ 8) (56-14)/ 8.  

2, -2, 2, -2, ...

This is a geometric progression.

First term = 2

The rate of geometric progression = -1

a1 = 2

a2 = a1 × (-1) = -2

a3 = a2 × (-1) = 2

And so on

⇒ This is a infinite geometric sequence

Answer:

Infinite geometric sequence

Step-by-step explanation:

2, -2, 2, -2, ...

Lets find the difference of the terms

-2 -2=0

2-(-2)=0

LEts check with common ratio

-2/2= -1

2/-2=-1

so common ratio r=0, so its geometric

The sequence is repeating because of common ratio -1

So it goes on infinitely

Hence it is Infinite geometric sequence

Answer:

1. 4n^3

2. 4k^7

3. 3

4. -30x

5. -6

Step-by-step explanation:

1. The prime factorization of 12 is 2 x 2 x 3 and the prime factorization of 16 is 2 x 2 x 2 x 2. When you look at these two expressions you can see the common factors of these two numbers are 2 x 2, which is 4. Next, we look at the GCF of the N's which would be n^3 since n^5 has three N's in it. Therefore, we get 4n^3 when we multiply the two together.

2. The factors of 8 are 1, 2, 4, and 8. Out of these, 1, 2, and 4 are the only factors that 20 shares with it and 4 is the greatest. Then, we look at the K's and the GCF of the K's is k^7 since k^8 has seven K's. We multiply the two and we get 4k^7.

3. Since one of the numbers of the three given here does not include the variable n, there will not be any N's in the GCF of the three, so we don't have to worry about that. Now, we just find the GCF of 18, -24, and -21. The factors of 18 are 1, 2, 3, 6, 9, and 18, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24, and lastly, the factors of 21 are 1, 3, 7, and 21. From these, 3 is the biggest common divisor, therefore the GCF is 3.

4. Between the two X's, X^1 is the biggest amount of X's this GCF has, so the final GCF will be some constant multiplies with X. Since we are dealing with bigger numbers on this problem, we should use prime factorization. The prime factorization of 90 is 2 x 3 x 3 x 5, and the prime factorization of 120 is 2 x 2 x 2 x 3 x 5. From these expressions, we take the biggest amount of each common factor as we can. Since these expressions both have 2, we take the smaller amount of 2's which is one two. Then we get one three from both expressions, and one five as well. 2 times 3 times 5 equals 30, therefore, we get -30x, and not 30x, because both of these numbers are negatives.

5. All of these numbers do not have an x, so there won't be an x in our GCF. Another method of quickly finding the GCF of numbers is to look at the smallest number's factors first to see what factors it shares with the other numbers. The factors of 12 are 1, 2, 3, 4, 6, and 12. 42 and 30 do not have the factor 12, so we can go down the list and see if 42 and 30 share the factor 6, which they do since 6 times 7 is 42 and 6 times 5 is 30. Since all of these numbers share the negative sign, the GCF of these three numbers is -6.  

The net of a cylinder is best described by a circle and one rectangle.

A cylinder is a three-dimensional solid, the most basics of curvilinear shapes which is considered as a prism with circle as its base.

A cylinder has a base radius and the height from its base to top .

Formula of surface area of cylinder is =  2πr(r + h)

Formula of Volume of cylinder is = [tex]\pi r^{2} h[/tex]

The net of the cylinder should have one side open such that it can be inserted within the cylinder.

As the top of the cylinder is circle, thus the net should have one circular top . Also the body of the cylinder is in the form of a rectangle which ensures the net should have also one rectangular body.

Therefore the net of a cylinder is best described by a circle and one rectangle.

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The net of a cylinder is comprised of 'one rectangle and two circles', which represent the lateral surface and the two equal-sized circular bases of the cylinder, respectively.

The net of a cylinder consists of two equal-sized circles and one rectangle that wraps around to form the curved surface. The two circles represent the top and bottom (or base) of the cylinder, and they are identical in size because the top and the bottom of a cylinder have the same cross-sectional area. The rectangle represents the lateral surface area of the cylinder, which, if 'unrolled', resembles a rectangle whose length is equal to the circumference of the circles (the perimeter of the base) and whose height is equal to that of the cylinder. The correct option that describes the net of a cylinder is thus 'one rectangle, two circles'.

Answer:2006

Step-by-step explanation:

[tex]A = 118e^{0.024t}[/tex]

When A = 140:

[tex]140 = 118e^{0.024t}[/tex]

[tex]\frac{140}{118} = e^{0.024t}[/tex]

[tex]ln(\frac{140}{118}) = 0.024t[/tex]

[tex]\frac{1}{0.024} ln(\frac{140}{118}) = t[/tex]

Plugging into a calculator, t is approximately 7.12.  Since t represents years since 1998, we round up to the nearest whole number: t=8.  So the population of the city will reach 140 thousand in the year 2006.

The population of the city reach 140 thousand will be after 7.123 years.

Let a be the initial value and x be the power of the exponent function and b be the increasing factor.

The exponent is given as

y = a(b)ˣ

The equation models the number of inhabitants in a specific city, in thousands, t years after 1998 is given below.

[tex]\rm A = 118 \times e^{0.024 \times t}[/tex]

The number of years when the population becomes 140 thousands is given as,

[tex]\rm 140 = 118 \times e^{0.024 \times t}[/tex]

Take natural log on both sides, then we have

0.024 t = ln (140 / 118)

0.024 t = 0.170957

t = 7.123 years

The population of the city reach 140 thousand will be after 7.123 years.

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[tex]\displaystyle\bf\\m \overset{\frown}{HE}=360-m \overset{\frown}{HL}-m \overset{\frown}{EV}-m \overset{\frown}{VL}\\m\overset{\frown}{HE}=360^o-40^o-130^o-110^o=360^o-280^o=80^o\\\\m\widehat{EYH}=m\widehat{EYV}=\frac{m \overset{\frown}{EV}-m\overset{\frown}{HE}}{2}=\frac{130^o-80}{2}=\frac{50^o}{2}=\boxed{\bf25^o}[/tex]

Answer:

  y = -(1/3)x -2

Step-by-step explanation:

For each horizontal "run" of 3 units, the "rise" of the line is -1 unit. Hence the slope is ...

   rise/run = -1/3

The y-intercept is where the line crosses the y-axis, at y = -2. So, the slope-intercept form of the equation of the line is ...

  y = (slope)·x + (y-intercept)

  y = -1/3x -2

1962 - 2003 = 41 years

In 2003 it’s value increased to = $435,000

$435,000 / 41 years

Per year’s value = $10,609.7561

B. 1950 - 1960 = 12 years

$60,000 / 12 years = $5000

Value of the house @ 1950 = $5000

Using proportions, it is found that:

Item a:

From an initial value of $60,000, the house increased in value by $375,000, as 435000 - 60000 = 375000.

The percent increase is given by:

[tex]\frac{375000}{60000} \times 100\% = 625\%[/tex]

In 2003 - 1962 = 41 years, hence:

[tex]r = \frac{625}{41} = 15.24[/tex]

The annual rate of increase in the value of the house was of 15.24%.

Item b:

The value increases 15.24% a year, hence, in t years after 1962, considering an initial value of $60,000, the value is:

[tex]V(t) = 60000(1.1524)^t[/tex]

1950 is 12 years before 1950, hence the value is V(-12), that is:

[tex]V(-12) = 60000(1.1524)^{-12} = \frac{60000}{(1.1524)^{12}} = 4029[/tex]

In 1950, the house was valued at $4,029.

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Answer:

1. Range =30

2. Variance =137.6

3. Standard deviation=11.7303

Step-by-step explanation:

This question requires you to find the range, variance and standard deviation of sample data set.

Given the data as; 127 150 121 120 140 128

Arrange the data in ascending order;

sample set S={120, 121, 127, 128, 140, 150}

number of elements, n=6

1. Range = Maximum (S) - Minimum (S) = 150- 120 = 30

⇒Find the mean of the data set

[tex]mean= \frac{120+121+127+128+140+150}{6} = 786/6 = 131[/tex]

2. Variance is the measure of how far a set of data is spread out.Standard deviation is the square-root of variance.To find variance you need to follow the steps below;

Finding the deviations from the mean and their squares

Deviations             Squares of deviations

120-131= -11               -11²=   121

121-131= -10                -10² =100

127-131= -4                  -4² = 16

128-131= -3                   -3= 9

140-131= 9                    9²=  81

150-131= 19                19²=  361

Finding the sum of the squares of the deviations from the mean

[tex]=121+100+16+9+81+361=688[/tex]

Finding the variance

Variance, S²=(sum of squares of deviations from mean)/ n-1

[tex]=\frac{688}{n-1} =\frac{688}{6-1} =\frac{688}{5} =137.6[/tex]

Finding standard deviation

Standard deviation , s , is the square-root of the variance 

[tex]s=\sqrt{137.6} =11.73[/tex]

Final Answer:

- Range: 30 mmHg
- Variance: 137.6 (mmHg)²
- Standard Deviation: Approximately 11.73 mmHg
- Ideal Standard Deviation: 0 mmHg

Explanation:

To find the range, variance, and standard deviation for the given blood pressure readings, we can follow these steps:
1. **Range:**
  - The range is the difference between the highest and lowest values in the data set.
  - Highest reading = 150 mmHg
  - Lowest reading = 120 mmHg
  - Range = Highest reading - Lowest reading = 150 - 120 = 30 mmHg

2. **Variance:**
  - Variance measures the average degree to which each reading differs from the mean of the readings. Because we are dealing with a sample of the population, not the entire population, we'll use the sample variance formula.
  - First, compute the mean of the readings.
  - Mean (average) blood pressure reading = (127 + 150 + 121 + 120 + 140 + 128) / 6
  - Mean = 786 / 6 = 131 mmHg
  - Now, we'll calculate the square of the differences between each reading and the mean, sum those, and divide by (n-1), where n is the number of readings.
  - Differences squared: (127-131)², (150-131)², (121-131)², (120-131)², (140-131)², (128-131)²
  - = (-4)², (19)², (-10)², (-11)², (9)², (-3)²
  - = 16, 361, 100, 121, 81, 9
  - Sum of squared differences = 16 + 361 + 100 + 121 + 81 + 9 = 688
  - Sample variance = 688 / (6 - 1) = 688 / 5 = 137.6 (mmHg)²

3. **Standard Deviation:**
  - The standard deviation is the square root of the variance and provides a measure of the average distance from the mean.
  - Standard deviation = √variance = √137.6 ≈ 11.73 mmHg

4. **Ideal Standard Deviation:**
  - If the subject's blood pressure remains constant, and the measurement technique is applied correctly and without any error, the ideal standard deviation should be zero because all measurements would be the same, resulting in no variability.

In summary:
- Range: 30 mmHg
- Variance: 137.6 (mmHg)²
- Standard Deviation: Approximately 11.73 mmHg
- Ideal Standard Deviation: 0 mmHg

Start by describing the data given about the flower Heliconia's varieties. Then, conduct a significance test, possibly using ANOVA, to compare their mean lengths. However, the computation for the p-value and F-statistical is not specified in the question. You can just round your numbers according to the instructions.

The first step in a complete analysis is the description of the data. From the question, we have three types of tropical flower Heliconia, namely H. bihai, H. caribaea red, and H. caribaea yellow. We have the respective data points for each class, n, the sample mean, x-bar, standard deviation, s, and the standard error, s_(x-bar).

The next step is to carry out a significance test. This can be done using ANOVA (Analysis of variance), which compares the means of three or more samples. The test statistic in ANOVA is the F statistic, and the null hypothesis is that the population means are equal.

Given numbers, you can compute the F-statistic, but from the question, it's unclear how the actual computation was done. The p-value can also be calculated from the F statistic; it's the probability of getting an extreme or more extreme result in your observed data, assuming the null hypothesis is true.

The instruction is explicit regarding how to round your numbers: sample mean to four decimal places, standard deviation to three decimal places, and your test statistic (F-statistic) to two decimal places. The p-value should also be rounded to three decimal places.

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[tex]f(x)=e^{-5x}[/tex] is continuous on [0, 1] and differentiable on (0, 1), so yes, the MVT is satisfied.

By the MVT, there is some [tex]c\in(0,1)[/tex] such that

[tex]f'(c)=\dfrac{f(1)-f(0)}{1-0}[/tex]

The derivative is

[tex]f'(x)=-5e^{-5x}[/tex]

so we get

[tex]-5e^{-5c}=e^{-5}-1\implies e^{-5c}=\dfrac{1-e^{-5}}5\implies-5c=\ln\dfrac{1-e^{-5}}5[/tex]

[tex]\implies\boxed{c=-\dfrac15\ln\dfrac{1-e^{-5}}5}[/tex]

The function f(x) = e^-5x is both continuous and differentiable on the interval [0, 1] and performs according to the Mean Value Theorem. To find the specific numbers, c, that suit the theorem’s conclusion, we must solve the equation f'(c) = [f(b) - f(a)] / (b - a).

The function we are considering is f(x) = e-5x. To check whether it satisfies the Mean Value Theorem (MVT) on the interval [0, 1], we have to ensure two conditions. Firstly, that the function is continuous on the closed interval [0, 1], and secondly, that it is differentiable on the open interval (0, 1).

Given that f(x) = e-5x is an exponential function, it is continuous and differentiable for all x in real numbers, R. Hence, f(x) is continuous and differentiable on [0, 1] and (0, 1), respectively. Therefore, the function satisfies the hypotheses of the Mean Value Theorem.

To find all the numbers c that satisfy the conclusion of the MVT, we have to solve the equation f'(c) = [f(b) - f(a)] / (b - a). Differentiating f(x), we get f'(x) = -5e-5x. On solving this equation for c, the value that satisfies it will be our solution.

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Answer:

i couldnt answer please be more specific

Step-by-step explanation:

namely, how many times does 2/7 go into 7/8?

[tex]\bf \cfrac{7}{8}\div\cfrac{2}{7}\implies \cfrac{7}{8}\cdot \cfrac{7}{2}\implies \cfrac{49}{16}\implies 3\frac{1}{16}\impliedby \textit{3 whole times}[/tex]

A 7/8 inch long string can be cut into 3 pieces of length 2/7 inch each.

This is an example of fraction division, which is related to Mathematics. To find out, how many pieces of string that are 2/7 of an inch long can be cut from a piece of string that is 7/8 of an inch long, you would have to divide the whole length of the string (7/8 inch) by the length of each piece (2/7 inch).

When you divide fractions, you actually multiply by the reciprocal of the second fraction. The reciprocal of a fraction is simply, flipping the numerator and denominator. So, the reciprocal of 2/7 would be 7/2.

Now simply multiply the two fractions, (7/8) times (7/2) which equals 49/16 or roughly 3.06. However, since you can't cut a string into a .06 piece, the answer would be 3 pieces.

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Answer:

Step-by-step explanation:

1. The function is good for years 1880 to 2009, 0 to 129 years after 1880. Values of x can be anything in the domain [0, 129].

__

2. 1950 -1880 = 70. The year 1950 corresponds to x=70, so the function tells us the water level rose ...

  f(70) = 1.6·70 = 112 . . . . . mm

__

3. We want to find x when f(x) = 100. That will be the solution to ...

  100 = 1.6x

  100/1.6 = x = 62.5

Then 62.5 years after 1880 is year 1942.5. The water level was 100 mm higher than in 1880 in the year 1942.

bank b for the loan and bank a for the savings account.

Answer:

Step-by-step explanation:

A) Let x represent acres of pumpkins, and y represent acres of corn. Here are the constraints:

  x ≥ 2y . . . . . pumpkin acres are at least twice corn acres

  x - y ≤ 10 . . . . the difference in acreage will not exceed 10

  12 ≤ x ≤ 18 . . . . pumpkin acres will be between 12 and 18

  0 ≤ y . . . . . the number of corn acres is non-negative

__

B) If we assume the objective is to maximize profit, the profit function we want to maximize is ...

  P = 360x +225y

__

C) see below for a graph

__

D) The profit for an acre of pumpkins is the highest, so the farmer should maximize that acreage. The constraint on the number of acres of pumpkins comes from the requirement that it not exceed 18 acres. Then additional profit is maximized by maximizing acres of corn, which can be at most half the number of acres of pumpkins, hence 9 acres.

So profit is maximized for 18 acres of pumpkins and 9 acres of corn.

Maximum profit is $360·18 +$225·9 = $8505.

-5

Make the equation into slope-intercept form, which is y = mx + b, where m is the slope, and b is the y-intercept.

In the equation y = -5x - 2/3, the slope is -5 and the y-intercept is -2/3.

For this case we have that the equation of the line of the slope-intersection form is given by:

[tex]y = mx + b[/tex]

Where:

m: It's the slope

b: It is the cutoff point with the y axis

We have the following equation:

[tex]y = - \frac {2} {3} -5x[/tex]

Reordering:

[tex]y = -5x- \frac {2} {3}[/tex]

So, we have to:

[tex]m = -5\\b = - \frac {2} {3}[/tex]

Answer:

The slope is -5

Answer:

20

Step-by-step explanation:

9/15 = 3/5

3*4=12

5*4=20

Answer:

20 shots

Step-by-step explanation:

First round

basket = 9

Total shots = 15

Percentage = 9/15 x 100 = 60%

Second round

baskets = 12

Total = x

(12/x) x 100 = 60%

12/x = 0.6

x = 12 ÷ 0.6

x = 20

Answer:

B.) Investing has the risk of losing principal, whereas saving does not.

Step-by-step explanation:

Saving can be accomplished a number of ways, including putting the money in a cookie jar (where it will not earn interest). Most savings institutions (banks, credit unions, and the like) are governed by rules that help to ensure the availability and safety of the balance. Often, such institutions are insured so that depositors are protected against loss of principal.

Many investment opportunities are governed by no such rules. The invested amount may be unavailable for perhaps a lengthy period of time, and any return on the investment may be dependent upon factors not under the control of the party accepting the money. There is the opportunity for complete loss of the invested amount, and the possibility of incurring additional liability in some cases.

Investment in certificates that are traded on a regulated exchange will be subject to the exchange rules, generally including the requirement that the investor be fully informed of the risks. That doesn't mean there is no risk—it just means the investor is supposed to be made aware of it.

Answer:

  8 ft

Step-by-step explanation:

The height of the cross section through the apex will be the same as the height of the apex: 8 ft.

Answer:

1. [tex]f(x)=-10^{x-1}-10[/tex] - [tex]-\frac{101}{10}[/tex]

2. [tex]f(x)=-3^{x+5}-9[/tex] - [tex]-252[/tex]

3. [tex]f(x)=-3^{x-2}-1[/tex] - [tex]-\frac{10}{9}[/tex]

4.  [tex]f(x)=-17^{x-1}+2[/tex] - [tex]\frac{33}{17}

Step-by-step explanation:

We are given the exponential functions and we are to match them with their y-intercepts.

1. [tex]f(x)=-10^{x-1}-10[/tex]:

Substituting x = 0 to find the y-intercept:

[tex]f(x)=-10^{0-1}-10 = -\frac{101}{10}[/tex]

y-intercept ---> [tex]-\frac{101}{10}[/tex]

2. [tex]f(x)=-3^{x+5}-9[/tex]:

Substituting x = 0 to find the y-intercept:

[tex]f(x)=-3^{x+5}-9=-252[/tex]

y-intercept ---> [tex]-252[/tex]

3. [tex]f(x)=-3^{x-2}-1[/tex]:

Substituting x = 0 to find the y-intercept:

[tex]f(x)=-3^{x-2}-1=-\frac{10}{9}[/tex]

y-intercept ---> [tex]-\frac{10}{9}[/tex]

4. [tex]f(x)=-17^{x-1}+2[/tex]:

Substituting x = 0 to find the y-intercept:

[tex]f(x)=-17^{x-1}+2=\frac{33}{17}[/tex]

y-intercept ---> [tex]\frac{33}{17}

Finding x.

Suzette ran = x

Suzette biked = 4x

7 = x + 4x --) 7 = 5x --)  1.4 = x

Suzette ran for 1.4 hours

Finding y.

Quickest way is to use Suzette biked = 4x and substitute the x for 1.4 to find how much she biked.

4 x 1.4 = 5.6

1.4 + 5.6 = 7 hours the total time exercising

And the mileage adds up to.

5(1.4) + 20(5.6) = 119

Answer:

  No

Step-by-step explanation:

The volume of a cube is the cube of the edge length, so the edge length of the cube-shaped block is ...

  edge length = ∛(64 cm³) = 4 cm

Then the smallest cross-section will be a square of edge length 4 cm, so will have an area of (4 cm)² = 16 cm².

The 16 cm² shape will not fit through an 8 cm² hole.

Using given area of the square hole, we find its side length to be approx. 2.83 cm. Calculating the side length of the block using its volume, we get 4 cm. As the block is larger than the hole, it won't fit.

The problem involves geometry, specifically the concepts of area and volume. The area of a square is given by the formula, A = s^2, where s is the side of the square. In this case, the area of the square hole is 8 sq cm, which means the side length of the square hole (s) is the square root of 8, or about 2.83 cm.

The volume of a cube is given by the formula V = s^3, where s is the side length of the cube. The volume of the cube block is 64 cubic cm, which means the side length of the block (s) is the cube root of 64, or 4 cm.

Therefore, since the side length of the block (4 cm) is greater than the side length of the square hole (2.83 cm), the block will not fit through the hole.

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Answer:

C. Domain: (negative infinity, infinity)   Range: (0, infinity)

Step-by-step explanation:

It's correct

The domain of the functions will be (-∞, ∞) and the range of the functions will be (0, ∞).

The domain means all the possible values of x and the range means all the possible values of y.

The functions are given below.

y = f(x)

y = g(x)

y = h(x)

y = k(x)

Then the domain of the functions will be (-∞, ∞) and the range of the functions will be (0, ∞).

Then the correct option is C.

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The answer is:

They will be 10 miles apart after 3 hours.

To calculate how long after they start will they be 10 miles apart, we need to assume that after 1 one hour, they were at the same distance (5 miles), then, calculate the time when they are 10 miles apart, knowing that Paul increased its speed two times, running first at 5mph and then, at 10 mph.

The time that will pass to be 10 miles apart can be calculated using the following equation:

[tex]TotalTime=TimeToReach5miles+TimeToBe10milesApart[/tex]

Calculating the time to reach 5 miles for both Larry and Paul, at a speed of 5 mph, we have:

[tex]x=xo+v*t\\\\5miles=0+5mph*t\\\\t=\frac{5miles}{5mph}=1hour[/tex]

We have that to reach a distance of 5 miles, they needed 1 hour. We need to remember that at this time, they were at the same distance.

If we want to know how many time will it take for them to be 10 miles apart with Paul increasing its speed to 10mph, we need to assume that after that time, the distance reached by Paul will be the distance reached by Larry plus 10 miles.

So, for the second moment (Paul increasing his speed) we have:

For Larry:

[tex]x_{L}=5miles+5mph*t[/tex]

Therefore, the distance of Paul will be equal to the distance of Larry plus 10 miles.

For Paul:

[tex]x{L}+10miles=xo+10mph*t\\\\5miles+5mph*t+10miles=5miles+10mph*t\\\\5miles+10miles-5miles=10mph*t-5mph*t\\\\10miles=5mph*t\\\\t=\frac{10miles}{5mph}=2hours[/tex]

Then, there will take 2 hours to Paul to be 10 miles apart from Larry after both were at 5 miles and Paul increased his speed to 10 mph.

Hence, calculating the total time, we have:

[tex]TotalTime=TimeToReach5miles+TimeToBe10milesApart[/tex]

[tex]TotalTime=1hour+2hours=3hours[/tex]

Have a nice day!

Answer:

The exact value of 2 sin(120°) cos(120°) is -√3/2

Step-by-step explanation:

* Lets revise the trigonometry functions of the double angle

# sin(2x) = 2 sin(x) cos(x)

# cos(2x) = cos²(x) - sin²(x) OR

  cos(2x) = 2 cos²(x) - 1 OR

  cos(2x) = 1 - 2 sin²(x)

# tan(2x) = 2 tan(x)/(1 - tan²(x))

* Now lets solve the problem

∵ 2 sin(120°) cos(120°)

- Put sin(120°) = sin(2×60°)

∵ sin(2x) = 2 sin(x) cos(x)

∴ sin(120°) = 2 sin(60°) cos(60°)

∵ sin(60°) = √3/2 and cos(60°) = 1/2

∴ sin(120°) = 2 (√3/2) (1/2) = √3/2

∴ sin(120°) = √3/2 ⇒ (1)

- Put cos(120°) = cos(2×60°)

∵ cos(2x) = cos²(x) - sin²(x)

∴ cos(120°) = cos²(60°) - sin²(60°)

∵ cos(60°) = 1/2 and sin(60°) = √3/2

∴ cos(120°) = (1/2)² - (√3/2)² = 1/4 - 3/4 = -2/4 = -1/2

∴ cos(120°) = -1/2 ⇒ (2)

- Substitute (1) and (2) in the expression 2 sin(120) cos(120)

∴ 2 sin(120°) cos(120°) = 2 (√3/2) (-1/2) = -√3/2

* The exact value of 2 sin(120°) cos(120°) is -√3/2

Answer:

in this problem we do a comparison case t i.e if 1/12 he writes 1/6 of page what about 1 page

     1/12minute  =   1/6

      ?    ×    1          then we cross multiply

(1*1/12)    ÷  1/6     =1/12*6  =  1/2 minute

The expression (5x + 2) + (5x + 2) + (5x + 2) simplifies to 15x + 6 by combining like terms; three 5x's give 15x, and three 2's give 6 when added together.

The expression (5x + 2) + (5x + 2) + (5x + 2) is given by adding three identical binomials. To find an equivalent expression, you can use the distributive property of multiplication over addition, which in this case can also be seen as simply combining like terms.

Step-by-step, here's how you simplify the expression:

So, (5x + 2) + (5x + 2) + (5x + 2) is equivalent to 15x + 6 for all values of x.

For this case we must indicate the percentage that represents the following expression:

[tex]\frac {1} {20}[/tex]

By a rule of three we can solve them:

20 ----------> 100%

1 ------------> x

Where the variable x represents the percentage of 1 with a base of 20.

[tex]x = \frac {1 * 100} {20}\\x = 5[/tex]

So, we have that [tex]\frac {1} {20}[/tex] represents 5%

Answer:

Option A

The required  1/20 is equal to 5% when expressed as a percentage. Option A is correct.

To find the percent equivalent to 1/20, we need to express it as a fraction of 100.

First, we can convert 1/20 into a decimal by dividing 1 by 20, which gives us 0.05.

Next, we multiply the decimal by 100 to express it as a percentage: 0.05 * 100 = 5%.

Therefore, 1/20 is equivalent to 5%.

In conclusion, 1/20 is equal to 5% when expressed as a percentage.

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The Lagrangian is

[tex]L(x,y,z,\lambda,\mu)=x^2+y^2+z^2+\lambda(4x^2+4y^2-z^2)+\mu(2x+4z-2)[/tex]

with partial derivatives (set equal to 0)

[tex]L_x=2x+8\lambda x+2\mu=0\implies x(1+4\lambda)+\mu=0[/tex]

[tex]L_y=2y+8\lambda y=0\implies y(1+4\lambda)=0[/tex]

[tex]L_z=2z-2\lambda z+4\mu=0\implies z(1-\lambda)+2\mu=0[/tex]

[tex]L_\lambda=4x^2+4y^2-z^2=0[/tex]

[tex]L_\mu=2x+4z-2=0\implies x+2z=1[/tex]

Case 1: If [tex]y=0[/tex], then

[tex]4x^2-z^2=0\implies4x^2=z^2\implies2|x|=|z|[/tex]

Then

[tex]x+2z=1\implies x=1-2z\implies2|1-2z|=|z|\implies z=\dfrac25\text{ or }z=\dfrac23[/tex]

[tex]\implies x=\dfrac15\text{ or }x=-\dfrac13[/tex]

So we have two critical points, [tex]\left(\dfrac15,0,\dfrac25\right)[/tex] and [tex]\left(-\dfrac13,0,\dfrac23\right)[/tex]

Case 2: If [tex]\lambda=-\dfrac14[/tex], then in the first equation we get

[tex]x(1+4\lambda)+\mu=\mu=0[/tex]

and from the third equation,

[tex]z(1-\lambda)+2\mu=\dfrac54z=0\implies z=0[/tex]

Then

[tex]x+2z=1\implies x=1[/tex]

[tex]4x^2+4y^2-z^2=0\implies1+y^2=0[/tex]

but there are no real solutions for [tex]y[/tex], so this case yields no additional critical points.

So at the two critical points we've found, we get extreme values of

[tex]f\left(\dfrac15,0,\dfrac25\right)=\dfrac15[/tex] (min)

and

[tex]f\left(-\dfrac13,0,\dfrac23\right)=\dfrac59[/tex] (max)

This problem involves using Lagrangian multipliers to optimize a function with two constraints. The maximum and minimum points can be found by solving the Lagrange equations, which are derivatives of the function and constraints. These points can be confirmed by checking the positive or negative value of the second-order derivative.

To find the maximum and minimum values of the function ​f(x,y,z)=x²+y²+z² subject the cone z²=4x²+4y² and the plane 2x+4z=2, we use Lagrange multipliers. We have two constraint functions here, given by the cone and the plane equations.

The first step is to set up the Lagrange equations, with and as Lagrange multipliers. From w=gradientf=gradientg+gradienth, we have three equations: 2x=*8x+2, 2y=*8y+0, 2z=*4. The second step is to solve these three equations, together with the original constraints ​g(x,y,z)=​0 and ​h(x,y,z)=​0.

Solving these equations will give you specific values for x, y, and z that correspond to the maximum and minimum points. To determine if a point is a maximum or minimum, one can compute the second-order partial derivatives and organize them into the Hessian matrix.

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