Answer :
The distance between the person and the shuttle is changing at a rate of 640 feet/sec when the shuttle is 4000 ft high and is rising at a rate of 800 ft/sec.
Considering x to be the height of the triangle and y to be the hypotenuse of the triangle, a right-angled triangle will be formed according to the given information having a base of 3000 feet which is the distance of the person from the launch pad.
In this right-angle triangle x will be the height of the shuttle and y will be the distance between the person and the shuttle.
Now we apply the Pythagorean theorem to the triangle;
y² = x²+(3000)²
Now differentiating this equation with respect to time,t :
2y (dy/dt) = 2x(dx/dt) + 0
dy/dt = x(dx/dt)/y
As the shuttle is rising at a speed of 800 ft/sec, (dx/dt)=800
Substitute 4000 for x into the equation, y²=x²+(3000)², to find y;
y² = (4000)²+(3000)²
y = 5000
Now we substitute 4000 for x, 5000 for y, and 800 for dx/dt in the equation, dy/dt = x(dx/dt)/y
dy/dt = 4000(800) / 5000
dy/dt = 640
Therefore, the distance between the person and the shuttle is increasing at a rate of 640 feet/sec.
To learn more about the Pythagorean theorem, click here:
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