Suppose that the distance an aircraft travels along a runway before takeoff is given by D = (5/9)t2, where D is measured in meters from the starting point and t is measured in seconds from the time the brakes are released. The aircraft will become airborne when its speed reaches 160 km/h. How long will it take to become airborne, and what distance will it travel in that time? How long will the airplane take to become airborne? sec (Type an integer or a decimal.)
Suppose that the distance an aircraft travels along a runway before takeoff is given by D = (5/9)t2, where D is measured in meters from the starting point and t is measured in seconds from the time the brakes are released. The aircraft will become airborne when its speed reaches 160 km/h. How long will it take to become airborne, and what distance will it travel in that time? How long will the airplane take to become airborne? sec (Type an integer or a decimal.)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem Description:**
Suppose that the distance an aircraft travels along a runway before takeoff is given by \( D = \left( \frac{5}{9} \right) t^2 \), where \( D \) is measured in meters from the starting point and \( t \) is measured in seconds from the time the brakes are released. The aircraft will become airborne when its speed reaches 160 km/h. How long will it take to become airborne, and what distance will it travel in that time?
**Question:**
How long will the airplane take to become airborne?
[Input box: ___] sec
(Type an integer or a decimal.)
**Explanation:**
To solve this problem, we need to determine the time \( t \) when the speed of the aircraft reaches 160 km/h.
1. **Convert the speed to meters per second**:
\[
160 \text{ km/h} = \frac{160 \times 1000}{3600} \text{ m/s} \approx 44.44 \text{ m/s}
\]
2. **Differentiate the distance function to find velocity \( v(t) \)**:
\[
D(t) = \left( \frac{5}{9} \right) t^2
\]
\[
v(t) = \frac{dD}{dt} = 2 \times \left( \frac{5}{9} \right) t = \frac{10}{9} t
\]
3. **Set the velocity equal to the speed**:
\[
\frac{10}{9} t = 44.44
\]
\[
t \approx \frac{44.44 \times 9}{10} \approx 40 \text{ sec}
\]
4. **Calculate the distance traveled in this time using \( D(t) \)**:
\[
D(40) = \left( \frac{5}{9} \right) \times 40^2 = \left( \frac{5}{9} \right) \times 1600 \approx 888.89 \text{ meters}
\]
**Conclusion:**
The airplane will take approximately 40 seconds to become airborne, covering a distance of about 888.89 meters.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F10b1f275-2c41-4985-b23a-aa3382c0d493%2F0374f38c-a4c8-448d-92ce-6592e4569c44%2F95eoi5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
Suppose that the distance an aircraft travels along a runway before takeoff is given by \( D = \left( \frac{5}{9} \right) t^2 \), where \( D \) is measured in meters from the starting point and \( t \) is measured in seconds from the time the brakes are released. The aircraft will become airborne when its speed reaches 160 km/h. How long will it take to become airborne, and what distance will it travel in that time?
**Question:**
How long will the airplane take to become airborne?
[Input box: ___] sec
(Type an integer or a decimal.)
**Explanation:**
To solve this problem, we need to determine the time \( t \) when the speed of the aircraft reaches 160 km/h.
1. **Convert the speed to meters per second**:
\[
160 \text{ km/h} = \frac{160 \times 1000}{3600} \text{ m/s} \approx 44.44 \text{ m/s}
\]
2. **Differentiate the distance function to find velocity \( v(t) \)**:
\[
D(t) = \left( \frac{5}{9} \right) t^2
\]
\[
v(t) = \frac{dD}{dt} = 2 \times \left( \frac{5}{9} \right) t = \frac{10}{9} t
\]
3. **Set the velocity equal to the speed**:
\[
\frac{10}{9} t = 44.44
\]
\[
t \approx \frac{44.44 \times 9}{10} \approx 40 \text{ sec}
\]
4. **Calculate the distance traveled in this time using \( D(t) \)**:
\[
D(40) = \left( \frac{5}{9} \right) \times 40^2 = \left( \frac{5}{9} \right) \times 1600 \approx 888.89 \text{ meters}
\]
**Conclusion:**
The airplane will take approximately 40 seconds to become airborne, covering a distance of about 888.89 meters.
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