Should the function that represents the amount of water pumped into the pool be a linear function or exponential function?
C
c
Use the functions you write in Parts A and B.
D
d
Draw the function C(t) and y=20 500 on the same axes.
A
a
See solution.
B
b
See solution.
C
c
See solution.
D
d
See solution.
Practice makes perfect
a
To write a function that represents the amount of water remaining after t hours, we will use an exponential decay function.
w(t)= a(1- r)^t
Here, a is the initial amount and r is rate of decay. We know the water evaporates at a rate of 0.5 % per hour, so r= 0.5 %= 0.005. Since we want to represent the amount of water after the pool has reached 19 000 gallons, a= 19 000.
w(t)= 19 000(1- 0.005)^t
⇕
w(t)=19 000(0.995)^t
b
When we turn on a hose, it will refill the pool at a rate of 300 gallons per hour. That is, the amount of water will increase 300 gallons each hour. Hence, the function p(t) represents the amount of water that is pumped into the pool.
p(t)=300* t
c
Let's add the functions in Parts A and B.
C(t)=& p(t) + w(t)
=&300t+ 19 000(0.995)^t
The function C(t) represents the amount of water remaining during the times in which more water is being pumped into the pool.
d
We want to determine how long will it take for the hose to fill the pool. To do so, we will use the graph of C(t) and the line y= 20 500.
We see that the functions intercept when t is about 7.2. Therefore, the hose must run about about 7.2 hours to fill the pool.
Showing Our Work
Graphing the function C(t)
To draw the graph of the function C(t), we will make a table of values.
x
300t+19 000(0.995)^t
C(t)=300t+19 000(0.995)^t
0
300* 0+19 000(0.995)^0
19 000
2
300* 2+19 000(0.995)^2
≈ 19 410
4
300* 4+19 000(0.995)^4
≈ 19 823
6
300* 6+19 000(0.995)^6
≈ 20 237
8
300* 8+19 000(0.995)^8
≈ 20 653
The ordered pairs ( 0, 19 000), ( 2, 19 410), ( 4, 19 823), ( 6, 20 237), and ( 8, 20 653) all lie on the graph of the function. Now, we will plot and connect these points with a smooth curve.
