|
Question 106443: Red candle: 45 cm tall and burns 3cm per hour
White candle: 30 cm tall and burns 1 1/2 cm per hour.
How tall will each candle be after 1 hour?
How tall will each candle be after 3 hours?
How tall will each candle be after 12 hours?
Can you generalize-a formula- a system for calculating the candle's height at any given time?
Which candle will last longer? How do you know?
If you burn both candles continously (same start time) will they ever be the same height at the same time?
Found 2 solutions by ankor@dixie-net.com, solver91311:
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! Red candle: 45 cm tall and burns 3cm per hour
White candle: 30 cm tall and burns 1 1/2 cm per hour.
:
How tall will each candle be after 1 hour?
Red: 45 - (1*3) = 42 cm
White: 30 - (1*1.5) = 28.5 cm
:
How tall will each candle be after 3 hours?
Red: 45 - (3*3) = 36 cm
White: 30 - (3*1.5) = 25.5 cm
:
How tall will each candle be after 12 hours?
Red: 45 - (12*3) = 9 cm
White: 30 - (12*1.5) = 12 cm
:
Can you generalize-a formula- a system for calculating the candle's height at any given time?
Let t = time in hrs and h = height in cm
Red: h = 45 - 3t
White: h = 30 - 1.5t
:
Which candle will last longer? How do you know?
The white one. The red burns twice as fast, but it is less than twice as tall
:
If you burn both candles continuously (same start time) will they ever be the same height at the same time?
:
Use the equations to see how long (t), they will be the same height
30 - 1.5t = 45 - 3t
+3t - 1.5t = 45 - 30
1.5t = 15
t = 15/1.5
t = 10 hrs they will be the same
:
Check the height when they are the same:
45 - (3*10) = 15"
30 - (1.5*10) = 15"
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website! If the first candle is 45cm tall, and burns at 3 cm per hour for 1 hour it should be:
45cm - (3cm)(1hr) = 42cm
And for the second candle:
30cm - (1.5cm)(1hr) = 28.5cm
Now let's look at the situation after 3 hours:
45cm - (3cm)(3hr) = 36cm
30cm - (1.5cm)(3hr) = 25.5cm
Are you beginning to see a pattern?
After 12 hours:
45cm - (3cm)(12hr) = 45 - 36 = 9cm
30cm - (1.5cm)(12hr) = 30 - 18 = 12cm
So how to generalize? Let's call this the height function with respect to time since the original height and the burn rate of a given candle are constants and the instantaneous height varies as to time.
h(t) = H - rt
Where h(t) is the height function with respect to time, H is the original height of the candle, r is the burn rate, and t is the time.
So how would you know how long a candle would last? Let's define the end of the candle's life as the time when the height is zero. So all we have to do is set h(t) = 0 and solve for t for a given candle's constant parameters and we should have a time value equal to how long the candle will last.
For the 45cm candle:
45 - 3t = 0
3t = 45
t = 15 hours
For the 30 cm candle:
30 - 1.5t = 0
1.5t = 30
t= 20 hours
and the 30 cm candle clearly lasts longer.
Here is a good place to check our answer. I like to look at my answer to a problem and ask myself, "Does that make sense?"
One of the candles burns twice as fast as the other, so in order for the two candles to last an equal amount of time, the faster burning one would have to start out twice as long. But 45cm is much less than twice as long as 30cm, so it makes sense that the 30cm candle would last longer.
The last part of the problem asks if the two candles would ever be the same height if you started them burning at the same time. That is the same thing as asking if the height functions, h(t) of each of the candles would ever be equal at a time in the interval 0 < t < 15 (remember 15 hours is the time the 45 cm candle burns out)
45 - 3t = 30 - 1.5t
45 - 30 = -1.5t + 3t
15 = 1.5t
t = 10
Since 10 is in the interval 0 < t < 15, we can say that the candles will be of equal height at t = 10 hours.
Check:
45 - 3(10) = 15
30 - 1.5(10) = 15
Now we know that both candles would be 15cm tall after they had burned for 10 hours, as well as knowing the "Yes" answer to the question is correct.
================================================================================
By the way, if you rearrange the height function formula we developed so that it looks like this: h(t) = -rt + H, do you see a similarity with the slope/intercept form of a straight line, y = mx + b? You should. If you graphed the two examples used in this problem using h(t) as the vertical axis and t on the horizontal axis, you would note that the line would cross the vertical axis (time 0) at a point equal to the original height of the candle and the steepness of the line would give you a graphical representation of how fast the candle was burning.
Noting where the line crossed the horizontal axis would give you the answer to the "how long would the candle last?" question, and noting where the two lines intersected would give you the answer to the "when are they the same height?" question.
|
|
|
| |
