The Clearwater Chemical Company is developing a product that uses enzymes to safely break down the foamy pollution from certain chemical spills. In a test, the company used the new product on foam that initially had a height of 100millimeters. After one hour, the foam had shrunk to a height of 80millimeters.
Let t represent the number of hours since the start of the test. Assuming the foam's height decreased exponentially, which expression best models the foam's height over time?
If the foam's height continued to decrease exponentially, what was the height 5hours after the start of the test?
Round your answer to the nearest whole number.
millimeters
The expression that best models the foam's height over time is:
h(t) = a * (1-r)^t
where h(t) represents the height of the foam at time t, a represents the initial height of the foam, r represents the decay rate, and t represents the number of hours since the start of the test.
We are given that the initial height of the foam is 100 millimeters. Let's substitute this value into the equation:
h(t) = 100 * (1-r)^t
We are also given that after one hour, the foam had shrunk to a height of 80 millimeters. Let's substitute this value into the equation:
80 = 100 * (1-r)^1
Now we can solve for the decay rate (r):
(1-r) = 80/100
(1-r) = 0.8
1-r = 0.8
-r = 0.8 - 1
-r = -0.2
r = 0.2
Now we can rewrite the equation with the decay rate:
h(t) = 100 * (1-0.2)^t
To find the height of the foam 5 hours after the start of the test, we can substitute t = 5 into the equation:
h(5) = 100 * (1-0.2)^5
h(5) = 100 * 0.8^5
h(5) ≈ 100 * 0.32768
h(5) ≈ 32.768
Rounded to the nearest whole number, the height of the foam 5 hours after the start of the test is 33 millimeters.