When a skydiver jumps from an airplane, his downward velocity increases until the force of gravity matches air resistance. The velocity at which this occurs is known as the terminal velocity. It is the upper limit on the velocity a skydiver in free fall will attain (in a stable, spread position), and for a man of average size, its value is about 176 feet per second (or 120 miles per hour). A skydiver jumped from an airplane, and the difference D = D(t) between the terminal velocity and his downward velocity in feet per second was measured at 2-second intervals and recorded in the following table. t = seconds into free fall D = velocity difference 0 176.00 2 124.19 4 87.63 6 61.83 8 43.63 10 30.79 (a) Show that the data are exponential. (Round your answer to two decimal places.) The successive ratios in D values are always , so the data can be modeled with an exponential function. Find an exponential model for D. (Let t be the seconds into free fall and D be the velocity difference in feet per second. Round your parameters to two decimal places.) D = 1.76 × 1.52t D = 30.79 × 1.16t D = 87.35 × 0.99t D = 176.00 × 0.84t D = 180.00 × 0.75t (b) What is the percentage decay rate per second for the velocity difference of the skydiver? (Use the model found in part (a).) % Explain in practical terms what this number means. The difference between the terminal velocity and the skydiver's velocity by % each second. (c) Let V = V(t) be the skydiver's velocity t seconds into free fall. Find a formula for V. V = 176.00 − 1.76 × 1.52t V = 176.00 − 30.79 × 1.16t V = 176.00 − 87.35 × 0.99t V = 176.00 − 176.00 × 0.84t V = 176.00 − 180.00 × 0.75t (d) How long would it take the skydiver to reach 95% of terminal velocity? (Use the model found in part (c). Round your answer to two decimal places.) s

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...

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When a skydiver jumps from an airplane, his downward velocity increases until the force of gravity matches air resistance. The velocity at which this occurs is known as the terminal velocity. It is the upper limit on the velocity a skydiver in free fall will attain (in a stable, spread position), and for a man of average size, its value is about 176 feet per second (or 120 miles per hour). A skydiver jumped from an airplane, and the difference

D = D(t)

between the terminal velocity and his downward velocity in feet per second was measured at 2-second intervals and recorded in the following table.

t = seconds into free fall	D = velocity difference
0	176.00
2	124.19
4	87.63
6	61.83
8	43.63
10	30.79

(a) Show that the data are exponential. (Round your answer to two decimal places.)

The successive ratios in D values are always , so the data can be modeled with an exponential function.

Find an exponential model for D. (Let t be the seconds into free fall and D be the velocity difference in feet per second. Round your parameters to two decimal places.)

D = 1.76 × 1.52^t

D = 30.79 × 1.16^t

D = 87.35 × 0.99^t

D = 176.00 × 0.84^t

D = 180.00 × 0.75^t

(b) What is the percentage decay rate per second for the velocity difference of the skydiver? (Use the model found in part (a).)
%

Explain in practical terms what this number means.

The difference between the terminal velocity and the skydiver's velocity by % each second.

V = V(t)

be the skydiver's velocity t seconds into free fall. Find a formula for V.

V = 176.00 − 1.76 × 1.52^t

V = 176.00 − 30.79 × 1.16^t

V = 176.00 − 87.35 × 0.99^t

V = 176.00 − 176.00 × 0.84^t

V = 176.00 − 180.00 × 0.75^t

(d) How long would it take the skydiver to reach 95% of terminal velocity? (Use the model found in part (c). Round your answer to two decimal places.)
s

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