Shelley spent 17 minutes washing dishes. She spent 38 minutes cleaning her room. Explain how you can use mental math to find how long Shelley spent on the two tasks.

Volume = pi * r^2 * h 
h = 3r
pi * r^2 * (3r) = 24 pi
3r^3 = 24
r^3 = 8
r = 2

 Height = 3*2 = 6 units

To find the amount of salt when the tank contains exactly 40 gallons, create and solve differential equations for salt concentration and tank size over time. We find the tank size is 40 at 10 minutes, at which point there is approximately 14.2 lbs of salt.

To solve this, you need to understand that the total amount of salt at any time t is equal to the amount of salt coming in minus the amount of salt going out.

To begin with, the tank has 80 gal x 1/8 lb/gal = 10 lbs of salt.

The amount of salt coming in is 4 gal/min * 1 lb/gal = 4 lbs/min

The amount of salt going out depends on the concentration of the salt in the tank at that time. This is (4-8)(total salt/liters in tank at time t).

Setting up a differential equation and solving gives us an equation for salt concentration and volume at time t:

The equation for the tank size(in gallons) at time t (in minutes) is: tank size = 80 - 4t

The equation for the salt in tank at time t (in minutes) is: salt = 10 - 4t + 80e^-2t

When the tank size is exactly 40 gallons, tank size = 40 = 80 - 4t so t = 10 minutes

Plugging t = 10 into our equation for salt gives us: salt = 10 - 4*10 + 80e^-20 = approximately 14.2 lbs.

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Answer:

The statements A,C,F,G and H are true.

Step-by-step explanation:

It is given that the radius of circle before dilation is [tex]\frac{1}{2}ft[/tex] and the scale factor is 8.

The circumference of original circle is,

[tex]S_1=2\pi r[/tex]

[tex]S_1=2\pi \times \frac{1}{2}=\pi[/tex]

The area of original circle is,

[tex]A_1=\pi r^2[/tex]

[tex]A_1=\pi (\frac{1}{2})^2[/tex]

[tex]A_1=\frac{\pi}{4}[/tex]

The dilation by scale factor 8 means the radius of new circle is 8 times of the original circle.

[tex]r=8\times \frac{1}{2}[/tex]

Therefore the radius of new circle is 4 ft and the statement A is true.

The circumference of original circle is,

[tex]S_2=2\pi r[/tex]

[tex]S_2=2\pi \times 4=8\pi[/tex]

[tex]\frac{S_2}{S_1}=\frac{8\pi}{\pi} =8[/tex]

The new circumference will be 8 times the original circumference. The statement C is true.

The area of original circle is,

[tex]A_2=\pi r^2[/tex]

[tex]A_2=\pi (4)^2[/tex]

[tex]A_2=16\pi[/tex]

[tex]\frac{A_2}{A_1}=\frac{16\pi}{\frac{\pi}{4}}=64[/tex]

The new area will be 64 times the original area. Therefore statement F is true.

The new circumference will [tex]8\pi[/tex],The new area will be  [tex]16\pi[/tex] square feet.

Answer

Find out the what is the approximate length of the side adjacent to angle C .

To prove

As given

In a right triangle, angle C measures 40°.

The hypotenuse of the triangle is 10 inches long.

Than by using the trignometric identity

[tex]cos\angle C= \frac{Base}{Hypotenuse}\\cos\angle C= \frac{BC}{AC}[/tex]

As shown the diagram is given below

AC= 10 inches , ∠C = 40 °

cos 40 = 0.766 (approx)

Put in the above formula

0.766 × 10 = BC

7.66 = BC

7.7 inches (approx) = BC

Option (b) is correct .

The correct option is Option C [tex]\boxed{{\mathbf{7}}{\mathbf{.66 inches}}}[/tex] .

Further explanation:

The cosine ratio can be represented as,

  [tex]\cos \theta  = \frac{{{\text{base}}}}{{{\text{hypotenuse}}}}[/tex]

Here, base is the length of the side adjacent to angle [tex]\theta[/tex]  and hypotenuse is the longest side of the right angle triangle.

The length of side opposite to angle [tex]\theta[/tex]  is perpendicular that is used for the sine ratio.

Step by step explanation:

Step 1:

From the given information, the observed right angle is attached.

First find the hypotenuse and the base of the right angle triangle.

It can be seen from the attached figure that the side [tex]BC[/tex]  is adjacent to angle [tex]C[/tex]  and the side [tex]AC[/tex]  is the hypotenuse of triangle.

Thus, the [tex]{\text{base}}=BC[/tex]  and [tex]{\text{hypotenuse}}=10[/tex] .

Step 2:

We know that the cosine ratio is [tex]\cos \theta =\frac{{{\text{base}}}}{{{\text{hypotenuse}}}}[/tex] .

Therefore, it can be written as,

 [tex]\cos \theta=\frac{{BC}}{{AC}}[/tex]

Now substitute the value [tex]BC=x[/tex]  and [tex]{\text{AC}}=10[/tex]  in the cosine ratio.

[tex]\begin{aligned}\cos C&=\frac{x}{{10}}\\{\text{co}}s40&=\frac{x}{{10}}\\0.766&=\frac{x}{{10}}\\x&=7.66\\\end{aligned}[/tex]

Therefore, the approximate length of the side adjacent to angle [tex]C[/tex]  is [tex]7.7{\text{ inches}}[/tex]  .

Thus, option C is correct.

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Answer details:

Grade: High school

Subject: Mathematics

Chapter: Trigonometry

Keywords: Distance, Pythagoras theorem, base, perpendicular, hypotenuse, right angle triangle, units, squares, sum, cosine ratio, adjacent side to angle, opposite side to angle.

Answer:

Option A The graph in the attached figure

Step-by-step explanation:

Let

x-----> the number of ream of paper

y-----> the number of ink cartridge

we know that    

[tex]6x+18y\leq 252[/tex] ----> inequality that represent the possible combinations of paper and ink cartridges that Adam may buy

using a graphing tool

the solution is the triangular shaded area

see the attached figure

Answer:

The value of [tex]k(x)=\frac{5^{2x}-1}{5^x}[/tex]

Step-by-step explanation:

We have given two function [tex]g(x)=5^x\text{and}h(x)=5^{-x}[/tex]

We have to find k(x)=(g-h)(x)

[tex]k(x)=g(x)-h(x)[/tex]           (1)

We will substitute the values in equation (1) we will get

[tex]k(x)=5^x-(5^{-x})[/tex]

Now, open the parenthesis on right hand side of equation we will get

[tex]k(x)=5^x-5^{-x}[/tex]

Using [tex]x^{-a}=\frac{1}{x^a}[/tex]

[tex]k(x)=5^x-\frac{1}{5^x}[/tex]

Now, taking LCM which is [tex]5^x[/tex] we will get after simplification

[tex]k(x)=\frac{5^{2x}-1}{5^x}[/tex]

Hence, the value of [tex]k(x)=\frac{5^{2x}-1}{5^x}[/tex]

Final answer:

The first consecutive term of the arithmetic sequence that sums to 303 is 150. The second consecutive term is 153. We found this by setting up an equation for the sum of two consecutive terms and solving for the first term.

Explanation:

To find the two consecutive terms in the arithmetic sequence 3, 6, 9, 12, ... that sum up to 303, we first need to understand the properties of an arithmetic sequence. The given sequence has a common difference of 3 (that is, each term is 3 more than the previous term). Let's denote the first of these two consecutive terms as a. Therefore, the next term would be a + 3 (since the common difference is 3).

We are given that the sum of these two terms is 303, so we can write an equation:

a + (a + 3) = 303

Combining like terms, we get:

2a + 3 = 303

Subtracting 3 from both sides gives:

2a = 300

Dividing both sides by 2 gives:

a = 150

So, the first term is 150 and the second term, being a + 3, is 153.

The function that should be used  to represent the price of the medal x years after 2000 is [tex]f(x) = 120 (1.10)^x[/tex]

Given that,

Based on the above information, the calculation is as follows:

[tex]f(x) = P(1 + rate)^t[/tex]

Here P means $120

rate is 10%

T = x

So, it should be

[tex]f(x) = 120 (1.10)^x[/tex]

Therefore we can conclude that The function that should be used  to represent the price of the medal x years after 2000 is [tex]f(x) = 120 (1.10)^x[/tex]

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Answer: APEX- 20.42

Step-by-step explanation: Multiply 13 by 0.07, multiply 16 by 0.21, and so on. then add up all of the decimals and that is your answer

The equation of line passing through points (-2,7) and (3,2) is y = -x + 5. The slope, m, is -1. The y-intercept, b, is 5.

The subject matter here is finding the equation of a line in slope-intercept form, which is expressed as y = mx + b. Here, 'm' is the slope of the line and 'b' is the y-intercept. The slope, m, can be found using the formula: m = (y2 - y1)/(x2 - x1). Applying the coordinates given, (-2,7) and (3,2), we find the slope, m = (2 - 7) / (3 - (-2)) = -5 / 5 = -1.

Then, substituting m, x, and y into the equation, we get the y-intercept. Using the point (-2,7), we have: 7 = -1*-2 + b -> 7 = 2 + b -> b = 7 - 2 = 5. So the y-intercept, b, is 5. Therefore, the equation of the line in slope-intercept form is y = -x + 5.

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To find acceleration 'a' in the equation F = ma, divide both sides by mass 'm', resulting in the formula [tex]a =\frac{f}{m}[/tex]

To solve for a in terms of F and m from the equation F = ma,

where F represents force,

m represents mass,

and a represents acceleration,

[tex]f= m*a\\a =\frac{f}{m}[/tex]

This gives us the formula:

[tex]a =\frac{f}{m}[/tex]

This formula tells us that the acceleration of an object is equal to the force applied to it divided by its mass.

To find the number of 2n-digit integers where odd position digits are odd, and even position digits are even and positive, we calculate based on the choices for each position, giving us a result of (5^n) * (4^n).

We encounter a combinatory problem in working out how many 2n-digit positive integers can be formed if the digits in odd positions must be odd and the digits in even positions must be even. Before proceeding, it's important to grasp the concept of positional numbering, where the rightmost digit is considered at position 1, and the counting proceeds from right to left.

For a 2n-digit positive integer, i.e., an integer with an even number of digits, there will be n digits at odd positions and n digits at even positions. For the odd positions, the digits can be any one of the five odd integers (1, 3, 5, 7, 9) and for the even positions, the digits can be any one of the four even positive integers (2, 4, 6, 8) because 0 is excluded as the question mentions they should be positive.

Therefore, for each position, we have a choice of five odd integers or four even integers. Since there are n odd positions and n even positions, we end up with (5^n) * (4^n) total possibilities or combinations.

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The total number of valid 2n-digit positive integers, where digits in odd positions must be odd and digits in even positions must be even, is calculated using the formula 5ⁿ * 4ⁿ.

To solve this problem, we need to consider the constraints given: digits in odd positions must be odd and digits in even positions must be even. Let's break down the problem step-by-step:

Total combinations = (Number of choices for odd positions) n * (Number of choices for even positions) n = 5ⁿ * 4ⁿ

This is the formula to calculate the number of valid 2n-digit integers under the given constraints.

Final answer:

Each friend ate approximately 0.3636 of a pizza when the four pizzas were divided evenly among 11 friends.

Explanation:

The student's question involves dividing four pizzas evenly among 11 friends, so each person gets the same proportion of pizza. We need to convert this proportion into decimal form to answer the question.

To find the proportion of a pizza that each friend ate, we calculate 4 pizzas ÷ 11 friends. So, each friend ate ≈ 0.3636 of a pizza. We arrived at this by dividing 4 by 11, which yields a repeating decimal, so we round it to four decimal places to express it accurately.

This value represents the proportion of pizza each person ate when the pizzas were divided equally.

Answer: P(S\cup T)=72%

Step-by-step explanation:

We are given that S and T are mutually exclusive events.

Therefore, the intersection of both the events must be 0.

i.e. [tex]p(S\cap T)=0[/tex]

P (S) = 65% and P(T) = 7%

We know that P(S or T)=[tex]P(S\cup T)=P(S)+P(T)-P(S\cap T)[/tex]

[tex]\Rightarrow P(S\cup T)=65\%+7\%=72\%[/tex]

Hence, P(S\cup T)=72%

Answer:

0.604 on a p e x

Since a pelidisi below ( 100 ) is considered undernourished and a pelidisi greater than ( 100 ) indicates overfeeding, with a pelidisi of ( 114 ), this individual is considered to be overfed.

Let's denote the pelidisi as ( P ), the weight in grams as ( W ), and the sitting height in centimeters as ( H). According to the given information, the pelidisi varies directly as the cube root of the person's weight and inversely as the person's sitting height. This relationship can be expressed mathematically as:

[tex]\[ P = k \times \frac{\sqrt[3]{W}}{H} \][/tex]

where ( k ) is the constant of variation.

We are given that a person with a weight of ( 48,820 ) g and a sitting height of ( 78.7 ) cm has a pelidisi of ( 100 ). We can use this information to find the value of ( k ):

[tex]\[ 100 = k \times \frac{\sqrt[3]{48820}}{78.7} \][/tex]

Solving for \( k \):

[tex]\[ k = \frac{100 \times 78.7}{\sqrt[3]{48820}} \]\[ k \approx \frac{7870}{36} \approx 218.611 \][/tex]

Now that we have the value of \( k \), we can find the pelidisi for a person with a weight of \( 54,688 \) g and a sitting height of \( 72.6 \) cm:

[tex]\[ P = 218.611 \times \frac{\sqrt[3]{54688}}{72.6} \][/tex]

Calculating \( P \):

[tex]\[ P \approx 218.611 \times \frac{38}{72.6} \]\[ P \approx 218.611 \times 0.522 \]\[ P \approx 114.14 \][/tex]

Rounded to the nearest whole number, the pelidisi of a person with a weight of ( 54,688 ) g and a sitting height of ( 72.6 ) cm is ( 114 ).

Since a pelidisi below ( 100 ) is considered undernourished and a pelidisi greater than ( 100 ) indicates overfeeding, with a pelidisi of ( 114 ), this individual is considered to be overfed.

The linear equation formed is C = 5250 + 175S, where C is the total cost and S is the amount of sugar in tons. This expresses the cost for Sugar Sweet Company to transport its sugar to the market, beginning from a fixed cost of $5250 with an additional $175 charged per ton of sugar transported.

The question involves the creation of a linear equation that represents the total cost, C, of transporting sugar. We are told the initial cost of renting trucks is $5250 and there's an additional cost of $175 for each ton of sugar, S.

Therefore, we can write the equation as: C = 5250 + 175S.

To graph this equation, start at the point (0, 5250) on the y-axis which represents the initial cost. The slope of the line is 175, which means for each ton of sugar transported, the cost increases by $175. From the starting point, you can plot other points moving up vertically 175 units for each unit moved to the right horizontally.

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Answer:c

Step-by-step explanation:

(4,-6), because this point makes both the equations true.