Answer :
The rate of change in distance between the runner and the second base at the instant the runner is 60 ft away from first base is calculated to be
-13.31 ft/sec.
Pythagoras theorem-
In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides, according to Pythagoras's Theorem. The three sides are known as the Perpendicular, Base, and Hypotenuse. Due to its position opposite the 90° angle, the hypotenuse in this case is the longest side. When the positive integer sides of a right triangle are squared, the result is an equation which is also called Pythagorean triple.
Given that [tex]\frac{dx}{dt} = 24 ft/sec\\[/tex]
At any instant given value of x is 60 ft. At this value of x we need to find the value [tex]\frac{dr}{dt}[/tex]
Now using Pythagoras theorem
[tex]x^{2} +y^{2} =r^{2}[/tex]
when x=60 , r= 108.16 ft (by Pythagoras theorem)
On differentiation we get
[tex]2x\frac{dx}{dt} +0= 2r\frac{dr}{dt} \\[/tex]
2×(-24)×60=2×108.16×[tex]\frac{dr}{dt}[/tex]
[tex]\frac{dr}{dt} = -13.31 ft/sec[/tex]
We can learn more about Pythagorean theorem at:
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