Question

15 apply math models deandre has 200 square inches of plastic to make a cylindrical insert for a bird feeder. which design should he choose? explain.

Explanation:

We need to calculate the surface area of each cylindrical design (note: for a bird feeder insert, we assume it is an open cylinder, so we calculate the lateral surface area plus one circular base area, as the top would be open for the feeder).

Step1: Define surface area formula

For a cylinder with radius $r$ and height $h$, the surface area (open top) is $SA = 2\pi r h + \pi r^2$.

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For Design 1 (top feeder):

Diameter = 4 in, so radius $r = \frac{4}{2} = 2$ in, height $h = 11$ in.

Step2: Calculate Design 1 SA

$$\begin{align*} SA_1 &= 2\pi (2)(11) + \pi (2)^2 \\ &= 44\pi + 4\pi \\ &= 48\pi \\ &\approx 48 \times 3.14 = 150.72 \text{ square inches} \end{align*}$$

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For Design 2 (bottom feeder):

Diameter = 9 in, so radius $r = \frac{9}{2} = 4.5$ in, height $h = 6$ in.

Step3: Calculate Design 2 SA

$$\begin{align*} SA_2 &= 2\pi (4.5)(6) + \pi (4.5)^2 \\ &= 54\pi + 20.25\pi \\ &= 74.25\pi \\ &\approx 74.25 \times 3.14 = 233.145 \text{ square inches} \end{align*}$$

Step4: Compare with available plastic

DeAndre has 200 square inches of plastic. $150.72 < 200$ and $233.145 > 200$.

Answer:

DeAndre should choose the first cylindrical design (4 in diameter, 11 in height). Its surface area is approximately 150.72 square inches, which is less than the 200 square inches of plastic he has, while the second design requires ~233.15 square inches, which exceeds his available plastic.