Question

practice 1
a baker makes muffins that serve 1 person each. for a party, the baker is asked to make a large muffin in the same shape as the individual muffins that will serve 100 people.
a. by what scale factor will the muffin need to be dilated? give the exact answer or round to the nearest tenth.
type your answers in the boxes.

b. the muffins are contained in decorative paper liners. how many times more paper will be required for the dilated muffin as for the original? give the exact answer or round to the nearest tenth.

Explanation:

Part (a)

Step 1: Relate volume to scale factor

The volume of a 3D object scales with the cube of the scale factor \( k \). The original muffin serves 1 person (volume \( V_1 \)), the dilated serves 100 (volume \( V_2 = 100V_1 \)). So \( V_2 = k^3V_1 \), so \( k^3=\frac{V_2}{V_1}=100 \).

Step 2: Solve for \( k \)

Take the cube root: \( k = \sqrt[3]{100} \approx 4.6 \) (since \( 4.6^3 = 4.6\times4.6\times4.6 = 21.16\times4.6 = 97.336 \), \( 4.7^3 = 4.7\times4.7\times4.7 = 22.09\times4.7 = 103.823 \), so closer to 4.6).

Step 1: Relate surface area to scale factor

Surface area of a 3D object scales with the square of the scale factor \( k \). We found \( k = \sqrt[3]{100} \), so surface area scale factor is \( k^2 = (\sqrt[3]{100})^2 = 100^{\frac{2}{3}}=\sqrt[3]{100^2}=\sqrt[3]{10000} \approx 21.5 \) (since \( 21.5^3 = 21.5\times21.5\times21.5 = 462.25\times21.5 = 9938.375 \), \( 21.6^3 = 21.6\times21.6\times21.6 = 466.56\times21.6 = 10077.696 \), so closer to 21.5).

Answer:

\( \sqrt[3]{100} \) (or approximately \( 4.6 \))

Part (b)