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 π. 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.

Explanation:

Part A

1. First, find the volume of the hemisphere. The formula for the volume of a hemisphere is \( V=\frac{2}{3}\pi r^{3} \). Given \( r = 1.5\space cm \) and \( \pi\approx3.14 \).
2. Calculate the volume: \( V=\frac{2}{3}\times3.14\times(1.5)^{3}=\frac{2}{3}\times3.14\times3.375 = 6.75\times3.14\times\frac{2}{3}= 6.75\times2.093\approx6.75\times2.093 = 14.13\space cm^{3}\) (wait, correction: \( \frac{2}{3}\times3.14\times3.375=\frac{2\times3.14\times3.375}{3}= 2\times3.14\times1.125 = 7.065\space cm^{3}\)).
3. For Plastic A: Mass = density \( \times \) volume = \( 1.3\times7.065 = 9.1845\space grams \). Cost = \( 9.1845\times0.09\approx0.8266\space\$ \).
4. For Plastic B: Mass = \( 1.0\times7.065 = 7.065\space grams \). Cost = \( 7.065\times0.12 = 0.8478\space\$ \).
5. Compare the two costs: \( 0.8266<0.8478 \), so Plastic A is cheaper.

Step1: Calculate the volume of the hemisphere.

The formula for the volume of a hemisphere is \( V=\frac{2}{3}\pi r^{3} \). Given \( r = 1.5\space cm \) and \( \pi = 3.14 \).
\[

$$\begin{align*} V&=\frac{2}{3}\times3.14\times(1.5)^{3}\\ &=\frac{2}{3}\times3.14\times3.375\\ &= 2\times3.14\times1.125\\ &= 7.065\space cm^{3} \end{align*}$$

\]

Step2: Calculate the mass of Plastic A.

Density of Plastic A is \( 1.3\space g/cm^{3} \). Mass \( m_{A}=\text{density}\times\text{volume}=1.3\times7.065 = 9.1845\space g \)

Step3: Calculate the cost of Plastic A.

Cost per gram of Plastic A is \( \$0.09 \). Cost \( C_{A}=9.1845\times0.09 = 0.826605\space\$ \approx0.83\space\$ \) (rounded to nearest cent)

Answer:

A. Plastic A

Part B