The rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall is [tex]\frac{527}{24}ft^2/sec[/tex].
What is triangle?
Three edges and three vertices make up a triangle, which is a polygon. It is among the fundamental shapes in geometry. Triangle ABC refers to a triangle with the vertices A, B, and C. In Euclidean geometry, any three points that are not collinear determine a singular triangle and a singular plane at the same time.
Given:
dx / dt = 2 feet / sec
We have to find dA/dt when x = 7.
Since,
Area = 1/2 x (xy)
A = 1/2 x (xy)
Differentiating both sides with respect to t
[tex]\frac{dA}{dt} = \frac{1}{2}(\frac{dx}{dt}y+\frac{dy}{dt}x)\\[/tex] ..(1)
We have,
x^2 + y^2 = 25^2
plug x = 7
⇒ y = √(25^2 - 7^2)
y = √(625 - 49)
y = √576
y = 24
Now plug x = 7, y = 24, dx/dt = 2 and dy/dt = -7/12 in equation (1)
[tex]\frac{dA}{dt} = \frac{1}{2} (2(24) + (-\frac{7}{12})(7)) \\ \frac{dA}{dt} = \frac{527}{24} ft^2/sec[/tex]
Hence, the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall is [tex]\frac{527}{24}ft^2/sec[/tex].
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