Question

7 from unit 5, lesson 13 find the volume of a pyramid whose base is a square with sides of length 4 units and that has a height of 10 units. 8 from unit 5, lesson 8 a solid has volume 4 cubic units and surface area 10 square units. the solid is dilated, and the image has volume 108 cubic units. what is the surface area of the image?

Explanation:

Step1: Calculate base area

The base is a square, so area is side length squared.
$B = 4^2 = 16$ square units

Step2: Apply pyramid volume formula

Volume of pyramid is $\frac{1}{3}Bh$, substitute $B=16$, $h=10$.
$V = \frac{1}{3} \times 16 \times 10$

Step3: Compute the volume

Calculate the product and simplify.
$V = \frac{160}{3} \approx 53.33$
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Step1: Find dilation factor (volume)

Dilation factor $k$ for volume: $k^3 = \frac{\text{new volume}}{\text{original volume}}$.
$k^3 = \frac{108}{4} = 27$

Step2: Solve for dilation factor

Take cube root of both sides.
$k = \sqrt[3]{27} = 3$

Step3: Scale surface area by $k^2$

Surface area scales with $k^2$, original area = 10.
$\text{New surface area} = 10 \times 3^2 = 10 \times 9 = 90$

Answer:

1. The volume of the pyramid is $\frac{160}{3}$ (or approximately 53.33) cubic units.
2. The surface area of the dilated image is 90 square units.