The diameter of each successive layer of a wedding cake is 2/3 the previous layer. If the diameter of the first layer of a 5 layer cake is 15 inches, find the sum of the circumferences of all the layers.
So the text says the answer is $39.1\pi$. But that is the sum of the diameters of the cake. Can someone verify the answer key is wrong.
On
The answer key is correct.
The circumference of a layer with diameter $d$ is $C = \pi d$. Since the diameter of each successive layer is $2/3$ that of the previous layer and the first layer is $15~\text{in}$, the sum of the circumferences of the five layers is $$15\pi~\text{in}\left[1 + \frac{2}{3} + \left(\frac{2}{3}\right)^2 + \left(\frac{2}{3}\right)^3 + \left(\frac{2}{3}\right)^4\right]$$ Since the $n$th partial sum of a geometric series with initial term $a_1$ and common ratio $r$ is $$S_n = a_1 \frac{1 - r^n}{1 - r}$$ the sum of the circumferences is \begin{align*} 15\pi~\text{in}\left[1 + \frac{2}{3} + \left(\frac{2}{3}\right)^2 + \left(\frac{2}{3}\right)^3 + \left(\frac{2}{3}\right)^4\right] & = 15\pi \cdot \dfrac{1 - \left(\frac{2}{3}\right)^5}{1 - \frac{2}{3}}\\ & = 15\pi~\text{in} \cdot \dfrac{1 - \frac{32}{243}}{\frac{1}{3}}\\ & = 45\pi~\text{in} \cdot \dfrac{211}{243}\\ & = 5\pi~\text{in} \cdot \frac{211}{27}\\ & = \frac{1055\pi}{27}~\text{in}\\ & \approx 39.1\pi~\text{in} \end{align*}
Recall that the circumference of a circle of diameter $d$ is $d\pi$. From the information given, we have that sum of circumferences is \begin{align} 15\pi \left(1 + \frac{2}{3} + \left(\frac{2}{3}\right)^2 + \left(\frac{2}{3}\right)^3 + \left(\frac{2}{3}\right)^4\right) \approx 39.1 \pi. \end{align}