Question

a toy piece made from plastic is in the shape of a solid hemisphere with a radius of 1.5 centimeters. there are two types of plastic that can be used to make the hemisphere. plastic a contains 1.3 grams per cubic centimeter and costs $0.09 per gram. plastic b contains 1.0 gram per cubic centimeter and costs $0.12 per gram. part a which plastic would result in the lower cost to make the piece? use 3.14 as an approximation for \\(\pi\\). a. plastic a b. plastic b part b what is the cost, to the nearest cent, to make one piece using the cheaper plastic? enter the answer in the box. $\square$

Explanation:

Step1: Calculate hemisphere volume

The volume of a hemisphere is $\frac{2}{3}\pi r^3$. Substitute $r=1.5$, $\pi=3.14$:
$$V = \frac{2}{3} \times 3.14 \times (1.5)^3 = \frac{2}{3} \times 3.14 \times 3.375 = 7.065 \text{ cm}^3$$

Step2: Find mass of Plastic A

Mass = density × volume. Plastic A density = 1.3 g/cm³:
$$m_A = 1.3 \times 7.065 = 9.1845 \text{ grams}$$

Step3: Calculate cost for Plastic A

Cost = mass × cost per gram. Plastic A cost = $0.09/g:
$$C_A = 9.1845 \times 0.09 = 0.826605 \text{ dollars}$$

Step4: Find mass of Plastic B

Plastic B density = 1.0 g/cm³:
$$m_B = 1.0 \times 7.065 = 7.065 \text{ grams}$$

Step5: Calculate cost for Plastic B

Plastic B cost = $0.12/g:
$$C_B = 7.065 \times 0.12 = 0.8478 \text{ dollars}$$

Step6: Compare costs to find cheaper

Compare $C_A$ and $C_B$: $0.826605 < 0.8478$, so Plastic A is cheaper.

Step7: Round cheaper cost to nearest cent

Round $0.826605$ to the nearest cent: $\approx 0.83$

Answer:

Part A: A. Plastic A
Part B: $\$0.83$