Question

a candle manufacturer sells cylindrical candles in sets of three. each candle in the set is a different size. the smallest candle has a radius of 0.5 inches and a height of 3 inches. the other two candles are scaled versions of the smallest, with scale factors of 2 and 3. how much wax is needed to create one set of candles?
a. 27 π cubic inches
b. 36 π cubic inches
c. 53 π cubic inches
d. 86 π cubic inches
e. 98 π cubic inches

Explanation:

Step1: Find volume formula for cylinder

The volume formula for a cylinder is $V = \pi r^{2}h$.

Step2: Calculate volume of smallest candle

Given $r_1=0.5$ inches and $h_1 = 3$ inches. Substitute into the formula: $V_1=\pi\times(0.5)^{2}\times3=\frac{3}{4}\pi$ cubic - inches.

Step3: Calculate dimensions of second - scaled candle

The scale factor of the second candle is 2. So $r_2 = 2\times0.5=1$ inch and $h_2=2\times3 = 6$ inches. Then $V_2=\pi\times1^{2}\times6 = 6\pi$ cubic - inches.

Step4: Calculate dimensions of third - scaled candle

The scale factor of the third candle is 3. So $r_3=3\times0.5 = 1.5$ inches and $h_3=3\times3=9$ inches. Then $V_3=\pi\times(1.5)^{2}\times9=\frac{81}{4}\pi$ cubic - inches.

Step5: Calculate total volume of the set

$V = V_1+V_2+V_3=\frac{3}{4}\pi+6\pi+\frac{81}{4}\pi=\frac{3 + 24+81}{4}\pi=\frac{108}{4}\pi = 27\pi$ cubic - inches.

Answer:

A. 27 $\pi$ cubic inches