The given scenario might be easier if you decide to use the same radius for all 3 candles. You can also choose a a relation between their heights. Alternatively, choose the dimensions of two candles and then calculate what the volume of the third one has to be.
B
The profit is equal to sales revenue minus the cost of the goods sold.
A
Example Candles:
Candle
Radius (inches)
Height (inches)
A
2.5
5.4
B
2.5
3.6
C
2.5
1.8
B
Example Prices: $2.50, $5, and $7.50, see solution.
Practice makes perfect
We want to make cylindrical candles of 3 different sizes to sell. Let's call them A, B, and C. To make it a bit easier, we can make them all with the same radius and only vary their heights. Let's also make the heights of the larger candles 2 and 3 times the height of the smallest.
We need to make 8 candles of each size and we have a total of 1 cubic foot of candle wax to use to create them. This gives us an equation for the total volume.
\begin{gathered}
8V_\text{A}+8V_\text{B}+8V_\text{C} = 1 \text{ ft}^3
\end{gathered}Let's convert the total volume to cubic inches using a conversion factor.
1 ft^3* 1728 in^3/1 ft^3=1728 in^3
We can substitute the formula for the volume of a cylinder into our equation and solve for h. Remember, the height for the cylinders are 3h, 2h, and h.
The height of Candle C should be 1.8 inches. Remember that we cannot round up since it will make us go over the total amount of wax we have. Rounding down means we will not use all of the wax, which is fine. Using the height of Candle C, we can find the height of Candle A and B as well.
Candle
Radius (inches)
Height (inches)
Volume (cubic inches)
A
2.5
3*1.8= 5.4
π ( 2.5)^2 ( 5.4) ≈ 106
B
2.5
2*1.8= 3.6
π ( 2.5)^2 ( 3.6) ≈ 70.7
C
2.5
1.8
π ( 2.5)^2 ( 1.8) ≈ 35.3
Please remember that these are only example dimensions for the candles. There are many possible combinations of candles sizes that would fit the given requirements.
Now we want to decide on a price for each candle to make a profit of $100. The profit is equal to sales revenue minus the cost of the goods sold. We know the cost of the goods is $20, so we can calculate the total amount of sales revenue we need.
To make a $100 profit, we need to sell the candles for a total of $120. Since we decided on a relation between the heights of the candles, and they all have the same radius, the same relation is true for the volumes.
\begin{gathered}
V_\text{A}=\pi {\color{#0000FF}{r}}^2 {\color{#009600}{3h}} \quad \Leftrightarrow\quad 3(\pi {\color{#0000FF}{r}}^2 {\color{#009600}{h}})=3V_\text{C}\\
V_\text{B}=\pi {\color{#0000FF}{r}}^2 {\color{#009600}{2h}} \quad \Leftrightarrow\quad 2(\pi {\color{#0000FF}{r}}^2 {\color{#009600}{h}})=2V_\text{C}
\end{gathered}
We can use this to split the price of each candle based on their volume. Let's call the price of Candle C p. Then the price should be 2p for Candle B and 3p for Candle A. This gives us an equation for finding p.
The price for Candle C should be $2.50. We can calculate the price for Candle A and B as well.
Candle
Price
A
3*$2.50=$7.50
B
2*$2.50=$5
C
$2.50
Just as with the sizes of the candles, there are many combinations of prices that would fit the given requirements. This is just one example of prices.
