Question

1. a cantaloupe can be roughly modeled as a sphere. using this approach, the cantaloupe at meijer has a volume of ( 288pi ) ( \text{in}^3 ) and the cantaloupe at aldi has a surface area of ( 256pi ) ( \text{in}^2 ). which store’s cantaloupe has a bigger radius? (hint: the volume of a sphere is given by ( v = \frac{4}{3}pi r^3 ) and the surface area of a sphere is given by ( sa = 4pi r^2 ).)

1. the directions on a box of brownie mix call for a 9 in × 13 in pan for brownies that are ( 1\frac{1}{2} ) inches high. unfortunately, you only have a 9-in round baking pan or a 13-in round baking pan. which of your two circular pans should you choose for best results? explain.

Explanation:

Problem 6

Step1: Find radius of Meijer's cantaloupe

Given volume \( V = 288\pi \) and \( V=\frac{4}{3}\pi r^3 \).
Set \( \frac{4}{3}\pi r^3 = 288\pi \).
Divide both sides by \( \pi \): \( \frac{4}{3}r^3 = 288 \).
Multiply both sides by \( \frac{3}{4} \): \( r^3 = 288\times\frac{3}{4}=216 \).
Take cube root: \( r = \sqrt[3]{216}=6 \).

Step2: Find radius of Aldi's cantaloupe

Given surface area \( SA = 256\pi \) and \( SA = 4\pi r^2 \).
Set \( 4\pi r^2 = 256\pi \).
Divide both sides by \( \pi \): \( 4r^2 = 256 \).
Divide by 4: \( r^2 = 64 \).
Take square root: \( r = \sqrt{64}=8 \).

Step3: Compare radii

Meijer's radius: \( 6 \) in, Aldi's radius: \( 8 \) in.
Since \( 8 > 6 \), Aldi's cantaloupe has a bigger radius.

To determine the best pan, we compare the volumes (or base areas, since height is same) of the rectangular pan and the circular pans.

Step1: Area of rectangular pan

Rectangular pan: \( 9 \times 13 = 117 \) \( \text{in}^2 \).

Step2: Area of 9 - in round pan (radius \( r = \frac{9}{2}=4.5 \) in)

Area of circle: \( A=\pi r^2=\pi(4.5)^2 = 20.25\pi\approx63.62 \) \( \text{in}^2 \).

Step3: Area of 13 - in round pan (radius \( r=\frac{13}{2}=6.5 \) in)

Area of circle: \( A=\pi(6.5)^2 = 42.25\pi\approx132.73 \) \( \text{in}^2 \).

Step4: Compare areas

Rectangular area: \( 117 \) \( \text{in}^2 \).
13 - in round pan area (\( \approx132.73 \)) is closer to \( 117 \) than 9 - in round pan (\( \approx63.62 \)).
(Or, volume of rectangular pan: \( 9\times13\times1.5 = 175.5 \) \( \text{in}^3 \).
Volume of 9 - in pan: \( \pi(4.5)^2\times1.5\approx95.43 \) \( \text{in}^3 \).
Volume of 13 - in pan: \( \pi(6.5)^2\times1.5\approx199.1 \) \( \text{in}^3 \).
The 13 - in pan’s volume is closer to the required volume.)

Answer:

Aldi's cantaloupe has a bigger radius.

Problem 7