3.5 cm

Question

3.9 cm
4.5 cm
A metalworker is creating decorative points in the form of triangular pyramids to go on top of a fence she is building. She wants to know the surface area of each pyramid. Use the surface area formula to find the surface area of each pyramid based on the figure.
__cm^2

Answer 1

To find the surface area of a triangular pyramid (also known as a tetrahedron), we'll use the following formula for the surface area (SA):

\[ SA = \frac{1}{2} \cdot P \cdot l + B \]

where:

• \( P \) is the perimeter of the base triangle,
• \( l \) is the slant height of the pyramid,
• \( B \) is the area of the base triangle.

Assuming the triangular base has sides corresponding to the dimensions you've provided (3.5 cm, 3.9 cm, and 4.5 cm), we first need to find the perimeter and the area of the base.

1. Calculate the perimeter (P) of the base triangle:

\[ P = 3.5 , cm + 3.9 , cm + 4.5 , cm = 12.9 , cm \]

2. Calculate the area (B) of the triangle using Heron's formula. Heron's formula is given by:

\[ s = \frac{a + b + c}{2} \] \[ B = \sqrt{s(s-a)(s-b)(s-c)} \]

Where \( a = 3.5 , cm\), \( b = 3.9 , cm\), and \( c = 4.5 , cm\).

Calculating the semi-perimeter \( s \):

\[ s = \frac{3.5 + 3.9 + 4.5}{2} = \frac{12.9}{2} = 6.45 , cm \]

Now substituting to find the area \( B \):

\[ B = \sqrt{6.45(6.45 - 3.5)(6.45 - 3.9)(6.45 - 4.5)} \] \[ B = \sqrt{6.45(2.95)(2.55)(1.95)} \]

Calculating the individual terms:

• \( 6.45 - 3.5 = 2.95 \)
• \( 6.45 - 3.9 = 2.55 \)
• \( 6.45 - 4.5 = 1.95 \)

Calculating the product:

\[ 2.95 \times 2.55 \times 1.95 \approx 14.682375 \]

Calculating the area \( B \):

\[ B = \sqrt{6.45 \times 14.682375} \approx \sqrt{94.88389375} \approx 9.74 , cm^2 \]

3. Assuming the slant height (l) is given, you would add that information here. Let's assume \( l \) is representative and typical for a small pyramid.

For illustration, let's say \( l = \) 5 cm.

4. Lastly, plug the values into the surface area formula:

\[ SA = \frac{1}{2} \cdot 12.9 \cdot 5 + 9.74 \] \[ SA = 32.25 + 9.74 = 41.99 , cm^2 \]

Thus, the surface area of the triangular pyramid is approximately

\[ \text{Surface Area} \approx 41.99 , cm^2. \]

You can replace the slant height with the actual value if provided to find the precise surface area.

Answer 2

What is the length of a right rectangular prism with a volume of 90, width of 3, and height of 5? (1 point)
__

Answer 3

To find the length of a right rectangular prism, you can use the formula for the volume \( V \) of a rectangular prism:

\[ V = \text{length} \times \text{width} \times \text{height} \]

Given:

• Volume \( V = 90 \)
• Width \( W = 3 \)
• Height \( H = 5 \)

We can rearrange the formula to solve for the length (\( L \)):

\[ L = \frac{V}{W \times H} \]

Substituting the known values into the equation:

\[ L = \frac{90}{3 \times 5} \]

Calculating the denominator:

\[ 3 \times 5 = 15 \]

Now, substitute back into the equation:

\[ L = \frac{90}{15} = 6 \]

Thus, the length of the right rectangular prism is

\[ \boxed{6} , \text{units}. \]

Answer 4

A right rectangular prism has an area of 20 square units and a height of 3 units. What is the volume? (1 point)
__units cubed

Answer 5

To find the volume of a right rectangular prism, you can use the formula for volume \( V \):

\[ V = \text{Base Area} \times \text{Height} \]

Given:

• Base Area \( A = 20 \) square units
• Height \( H = 3 \) units

Now, substitute the values into the volume formula:

\[ V = 20 , \text{square units} \times 3 , \text{units} \]

Calculating the volume:

\[ V = 60 , \text{cubic units} \]

Thus, the volume of the right rectangular prism is

\[ \boxed{60} , \text{units cubed}. \]

Answer 6

A gift box is being constructed in the shape of a right rectangular prism. The base is 15 inches squared and the height is 5 inches. What will the volume of the gift box be? (1 point)
__in.3

Answer 7

To find the volume \( V \) of a right rectangular prism (like a gift box), you can use the formula:

\[ V = \text{Base Area} \times \text{Height} \]

Given:

• Base Area \( A = 15 \) square inches
• Height \( H = 5 \) inches

Now, substitute the values into the volume formula:

\[ V = 15 , \text{in}^2 \times 5 , \text{in} \]

Calculating the volume:

\[ V = 75 , \text{in}^3 \]

Thus, the volume of the gift box is

\[ \boxed{75} , \text{in}^3. \]

Answer 8

What is the volume of a rectangular pyramid with a length of 32 cm, a width of 25 cm, and an altitude of 41 cm? (1 point)
98, 400 cm'
• 10, 933.33 сm3
32.67 cm'
32, 800 cm'

Answer 9

To find the volume \( V \) of a rectangular pyramid, you can use the formula:

\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

The base area (\( A \)) of a rectangular pyramid can be calculated using:

\[ A = \text{length} \times \text{width} \]

Given:

• Length = 32 cm
• Width = 25 cm
• Height (altitude) = 41 cm

First, calculate the base area:

\[ A = 32 , \text{cm} \times 25 , \text{cm} = 800 , \text{cm}^2 \]

Now, substitute the base area and height into the volume formula:

\[ V = \frac{1}{3} \times 800 , \text{cm}^2 \times 41 , \text{cm} \]

Calculating the volume:

\[ V = \frac{1}{3} \times 800 \times 41 = \frac{32,800}{3} \approx 10,933.33 , \text{cm}^3 \]

Thus, the volume of the rectangular pyramid is

\[ \boxed{10,933.33 , \text{cm}^3}. \]

Answer 10

A resort is building a scuba diving lagoon in the shape of an upside down pyramid. The top of the lagoon is a rectangle that is 50 feet by 20 feet, and the deepest point of the lagoon is in the middle at 25 feet deep. How much water does the lagoon hold? (1 point)
O 75, 000 ft. 3
• 95 ft.3
0 8,333.33 ft. 3
O 25, 000 ft. 3

Answer 11

To find the volume of a pyramid-shaped lagoon (in this case, an upside-down pyramid), you can use the formula for the volume \( V \) of a pyramid:

\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

Given:

• The base of the lagoon is a rectangle with dimensions:
  • Length = 50 feet
  • Width = 20 feet
• The height (or depth, in this case) of the lagoon is 25 feet.

Step 1: Calculate the Base Area

The base area \( A \) of the rectangle is:

\[ A = \text{Length} \times \text{Width} = 50 , \text{ft} \times 20 , \text{ft} = 1000 , \text{ft}^2 \]

Step 2: Calculate the Volume

Now plug the base area and the height into the volume formula:

\[ V = \frac{1}{3} \times 1000 , \text{ft}^2 \times 25 , \text{ft} \]

Calculating the volume:

\[ V = \frac{1}{3} \times 1000 \times 25 = \frac{25000}{3} \approx 8333.33 , \text{ft}^3 \]

Thus, the lagoon holds approximately

\[ \boxed{8333.33 , \text{ft}^3}. \]

This option matches the choices provided.