Question

A candy bar box is in the shape of a triangular prism. The volume of the box is 3,240 cubic centimeters. A triangular prism is shown with base of triangle labeled 24 cm, sides of the triangle labeled 15 cm, and length of the box equal to 30 cm. Part A: What is the height of the base? Show your work. (5 points) Part B: What is the approximate amount of cardboard used to make the candy box? Explain how you got your answer.

Answer 1

A: The height of the base of the triangular prism is 9 cm.
B: Approximate amount of cardboard used to make the candy box is 1476 cm^2

Step-by-step explanation:

Part A: What is the height of the base?

To find the height of the base of the triangular prism, we can use the formula for the volume of a triangular prism:

`\[ V = (1)/(2) * \text{base length} * \text{height of the base} * \text{length of the prism} \]`

Given:
- Volume (V) of the prism = 3,240 cm³
- Base length of the triangle = 24 cm
- Length of the prism = 30 cm

Rearrange the formula for height (h) of the base triangle:

`\[ h = \frac{2 * V}{\text{base length} * \text{length of the prism}} \]`

Plug in the given values:

`\[ h = (2 * 3,240)/(24 * 30) \\\\\[ h = (6,480)/(720) \]`

Perform the division:
h = 9

So, the height of the base of the triangular prism is 9 cm.

Part B: What is the approximate amount of cardboard used to make the candy box?

The amount of cardboard used to make the box can be approximated by calculating the surface area of the triangular prism. A triangular prism has 5 faces: 2 triangular bases and 3 rectangular sides.

First, let's calculate the area of one of the triangular bases:

`\[ \text{Area of triangle} = (1)/(2) * \text{base length} * \text{height of the base} \\\\\[ \text{Area of triangle} = (1)/(2) * 24 * 9 \\\\\[ \text{Area of triangle} = 12 * 9 \\\\\[ \text{Area of triangle} = 108 \, \text{cm}^2 \]`

There are two triangles, so the total area of the triangular bases is:

`\[ 2 * \text{Area of triangle} \\\\\[= 2 * 108 \\\\\[= 216 \, \text{cm}^2 \]`

Next, let's calculate the area of the three rectangles. The length of each rectangle is equal to the length of the prism, and the width is equal to the side of the triangle.

For two rectangles, the width is equal to the height of the triangle we have found:

`\[ \text{Area of each rectangle} = \text{height of the base} * \text{length of the prism} \\\\\[ \text{Area of each rectangle} = 9 * 30 \\\\\[ \text{Area of each rectangle} = 270 \, \text{cm}^2 \]`

For one rectangle, the width is the base length of the triangle:

`\[ \text{Area of the rectangle} = \text{base length} * \text{length of the prism} \\\\\[ \text{Area of the rectangle} = 24 * 30 \\\\\[ \text{Area of the rectangle} = 720 \, \text{cm}^2 \]`

Adding the area of three rectangles:

`\[ 2 * 270 \, \text{cm}^2 + 720 \, \text{cm}^2 \\\\\[= 540 \, \text{cm}^2 + 720 \, \text{cm}^2 \\\\\[= 1,260 \, \text{cm}^2 \]`

To find the total surface area, add the areas of the triangles and rectangles together:

`\[ \text{Total Surface Area} = 216 \, \text{cm}^2 + 1,260 \, \text{cm}^2 \\\\\[ \text{Total Surface Area} = 1,476 \, \text{cm}^2 \]`

This is the approximate amount of cardboard used to make the candy box.

Answer 2

The answer to Part A is 9...or I'm pretty sure it is

Step-by-step explanation:

hb=2 * area/b

b=24

Area=108 cm ^2

hb= 2 * 108/24

hb= 2 * 4.5

hb=9 cm

The height of the base is equal to 9 cm.