Answer :
Height of the ladder is 13 m.
How to differentiate the function with two or more variables?
The partial derivatives of a function z=f(x,y) are dz/dx and dz/dy. These derivatives provide the instantaneous rates of change for each of the independent variables
Let us assume that the ground is horizontal and the ladder has a constant length.
Let, the heigth of the ladder on the wall = y
distance between the bottom of the wall and the bottom of the ladder = x
length of the ladder = h
As the ladder is sliding down the wall, so y is decreasing at the rate of 0.375 m/s
[tex]\frac{dy}{dt}[/tex] = -0.375 m/s
The bottom of the ladder is sliding away from the wall, so x is increasing at the rate of 0.9 m/s when x = 5 m
[tex]\frac{dx}{dt}[/tex] = 0.9 m/s when x = 5 m
By using pythagoras theorem,
[tex]x^{2} +y^{2} =h^{2}[/tex]
On differentiating both sides,
[tex]2x[/tex] [tex]\frac{dx}{dt}+2y\frac{dy}{dt} =0[/tex]
2(5)(0.9) + 2y(-0.375) = 0
9 - 0.75y = 0
9 = 0.75y
y = 9/0.75 = 12
When x = 3 , we have y = 12
Also, [tex]x^{2} +y^{2} =h^{2}[/tex]
[tex]5^{2}+12^{2} =h^{2}[/tex]
25 + 144 = [tex]h^{2}[/tex]
169 = [tex]h^{2}[/tex]
[tex]13^{2}=h^{2}[/tex]
h = 13
Therefore, height of the ladder is 13 m.
To learn more about the differentiation from the given link.
https://brainly.com/question/23819325
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