Suppose that an oil spill in a lake covers a circular area and that the radius of the circle is increasing according to the formula r = f(t) = 12 + t1.65, where t represents the number of hours since the spill was first observed and the radius r is measured in meters. (Thus when the spill was first discovered, t = 0 hr, and the initial radius was r = f(0) = 12 + 01.65 = 12 m.) (a) Let A(r) = πr2, as in Example 5. Compute a table of values for the composite function A ∘ f with t running from 0 to 5 in increments of 0.5. (Round each output to the nearest integer.) Then use the table to answer the questions that follow in parts (b) through (d). t 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 A (b) After one hour, what is the area of the spill (rounded to the nearest 10 m2)? m2 (c) Initially, what was the area of the spill (when t = 0)? m2 Approximately how many hours does it take for this area to double? hr (d) Compute the average rate of change of the area of the spill from t = 0 to t = 2.5. (Round your answer to one decimal place.) m2/hr Compute the average rate of change of the area of the spill from t = 2.5 to t = 5. (Round your answer to one decimal place.) m2/hr Over which of the two intervals is the area increasing faster? t = 0 to t = 2.5t = 2.5 to t = 5
Suppose that an oil spill in a lake covers a circular area and that the radius of the circle is increasing according to the formula r = f(t) = 12 + t1.65, where t represents the number of hours since the spill was first observed and the radius r is measured in meters. (Thus when the spill was first discovered, t = 0 hr, and the initial radius was r = f(0) = 12 + 01.65 = 12 m.) (a) Let A(r) = πr2, as in Example 5. Compute a table of values for the composite function A ∘ f with t running from 0 to 5 in increments of 0.5. (Round each output to the nearest integer.) Then use the table to answer the questions that follow in parts (b) through (d). t 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
| t | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 |
| A |
(b) After one hour, what is the area of the spill (rounded to the nearest 10 m2)?
m2
(c) Initially, what was the area of the spill (when t = 0)?
m2
Approximately how many hours does it take for this area to double?
hr
(d) Compute the average rate of change of the area of the spill from t = 0 to t = 2.5. (Round your answer to one decimal place.)
m2/hr
Compute the average rate of change of the area of the spill from t = 2.5 to t = 5. (Round your answer to one decimal place.)
m2/hr
Over which of the two intervals is the area increasing faster?
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