Respuesta :
Using the Pythagorean Theorem, it is found that the rate of change of the hypotenuse is of -9.8 in/sec.
What is the Pythagorean Theorem?
The Pythagorean Theorem relates the length of the legs [tex]l_1[/tex] and [tex]l_2[/tex] of a right triangle with the length of the hypotenuse h, according to the following equation:
[tex]h^2 = l_1^2 + l_2^2[/tex]
Applying implicit differentiation, the rate of change of the hypotenuse is given by:
[tex]2h\frac{dh}{dt} = 2l_1\frac{dl_1}{dt} + 2l_2\frac{dl_2}{dt}[/tex]
[tex]h\frac{dh}{dt} = l_1\frac{dl_1}{dt} + l_2\frac{dl_2}{dt}[/tex]
In this problem, we have that the parameters are given as follows:
[tex]l_1 = 21, \frac{dl_1}{dt} = -7, l_2 = 28, \frac{dl_2}{dt} = -7[/tex]
[tex]h = \sqrt{l_1^2 + l_2^2} = \sqrt{21^2 + 28^2} = 35[/tex]
Hence, the rate of change of the hypotenuse, in inches per second, is given by:
[tex]h\frac{dh}{dt} = l_1\frac{dl_1}{dt} + l_2\frac{dl_2}{dt}[/tex]
[tex]35\frac{dh}{dt} = 21(-7) + 28(-7)[/tex]
[tex]\frac{dh}{dt} = \frac{21(-7) + 28(-7)}{35}[/tex]
[tex]\frac{dh}{dt} = -9.8[/tex]
More can be learned about the Pythagorean Theorem at https://brainly.com/question/654982
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