Question

#2 a cylindrical cup is 8 centimeters in height. when filled to the very top, it holds approximately 480 cubic centimeters of water. what is the radius of the cup, rounded to the nearest tenth? show all your work. #3 a food company is designing containers for several products. each container is a cylinder. the company makes a single serving oatmeal container that holds 1.2 ounces of oatmeal. they plan to make an extra large container for school cafeterias. the extra large container will be a dilation of the single serving container using a scale factor of 4. how many ounces of oatmeal will the extra large container hold? show all your work.

Explanation:

Problem #2

Step 1: Recall the formula for the volume of a cylinder

The volume \( V \) of a cylinder is given by the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. We know \( V = 480 \) cubic centimeters and \( h = 8 \) centimeters. We need to solve for \( r \).

Step 2: Substitute the known values into the formula

Substitute \( V = 480 \) and \( h = 8 \) into the formula:
\( 480 = \pi r^2 \times 8 \)

Step 3: Solve for \( r^2 \)

First, divide both sides of the equation by \( 8\pi \):
\( r^2 = \frac{480}{8\pi} \)
Simplify the right - hand side:
\( r^2=\frac{60}{\pi} \)

Step 4: Solve for \( r \)

Take the square root of both sides:
\( r=\sqrt{\frac{60}{\pi}} \)
Calculate the value:
\( r\approx\sqrt{\frac{60}{3.14159}}\approx\sqrt{19.0986}\approx4.4 \) (rounded to the nearest tenth)

Step 1: Recall the relationship between the volume of similar solids and the scale factor

For similar solids, if the scale factor of the linear dimensions (radius and height in the case of a cylinder) is \( k \), the ratio of the volumes is \( k^3 \). Here, the scale factor \( k = 4 \) (since the extra - large container is a dilation of the single - serving container with a scale factor of 4).

Step 2: Calculate the volume of the extra - large container

The volume of the single - serving container \( V_1=1.2 \) ounces. Let the volume of the extra - large container be \( V_2 \).
We know that \( \frac{V_2}{V_1}=k^3 \), where \( k = 4 \). So \( V_2=V_1\times k^3 \)

Step 3: Substitute the values and calculate

Substitute \( V_1 = 1.2 \) ounces and \( k = 4 \) into the formula:
\( V_2=1.2\times4^3 \)
First, calculate \( 4^3=4\times4\times4 = 64 \)
Then, \( V_2=1.2\times64 = 76.8 \) ounces.

Answer:

The radius of the cup is approximately 4.4 centimeters.

Problem #3