2 candles in a supermarket

Question

In the supermarket we can buy two different types of candles.

· The first type is 20 𝑐𝑚 long when new and burns at a rate of 2.5 𝑚𝑚 per hour.

· The second type is 24.5 𝑐𝑚 long when new and burns at a rate of 4 𝑚𝑚 per hour. We shall use 𝑡 to denote the time in hours, ℎ1 the function that gives the height in 𝑚𝑚 of the first type of candles after 𝑡 hours and ℎ2 the function that gives the height in 𝑚𝑚 of the second type of candles after 𝑡 hours.

1. Find the formula for ℎ1 (𝑡) and for ℎ2 (𝑡)

2. How long will each candle last?

3. Draw the graph of ℎ1 and ℎ2 on the same set of axes. Don’t forget to explain how you did your drawing.

4. Suppose we light both candles at the same time. After how many hours will they have the same height? Justify.

 5. If you want the candles to end at the same time, which candle would you light first, and how long after this would you light the second one?

Answer 1

Hello, Jimenez,

Candle heights, H, are 20cm and 24.5 cm for the two candles, given by H1 and H2. We know their rates of burn are 2.5 and 4 mm/hr. Let's put the units on the same bases so as not to get confused. 1 cm = 10mm, so I'll convert the heights to 200 and 245mm, respectively.

The height as a function of time will be the starting height minus the rate at which the candle burns multiplied by the time, in hours.

Candle 1: H1(t) = 200mm - (2.5mm/hr)*t, where t is time, in hours. At time = 0, we get the original height. After 2 hours, we lose 5mm of height, and so on.

Candle 2: H2(t) = 245mm - (4mm/hr)*t. This candle starts at 245 mm (t=0) and loses 4mm for each hour it burns.

To find how long each candle will last, set their heights to zero.

Candle1: 0 = 200mm -(2.5mm/hr)*t

t = 80 hours

Candle 2: Do the same thing and we find the life of the candle is 61.25 hours.

To determine the time their heights are equal, set each equation to equal to the other. In other words H1 = H2.

200 - 2.5t = 245 - 4t

t = 45/1.5, or 30 hours. Make the graph and you'll see where the lines cross, at 30 hours. Y should be the height in cm, and X is the time in hours.

You want to light the slowest candle first. It takes 80 hours. Since candle 2 takes 61.25 hours, take the difference (80 - 61.25 = 18.75 hours) and that will be the time to wait before starting the second candle.

I hope this helps,

Bob