The base of an isosceles triangle is 10 feet long and the base angles are decreasing at a rate of 2° per second. Find the rate of change of the area when the base angles are 45°.
we have the height h is
h = 5tanθ
a = 5h = 25tanθ
da/dt = 25sec^2θ dθ/dt
Now just plug in your numbers (in radians!)
Thank you Sir ☺
But where did the a=5h come from?
did you actually draw an isosceles triangle with a base of 10?
A = 1/2 bh
Let's not forget the basics now that we're in calculus!
Oh yes. Sorry sir. i just got confused. Thank you :)
To find the rate of change of the area of the isosceles triangle, we need to use the chain rule in calculus.
Let's denote the base of the triangle as b, the base angles as A (both are functions of time t), and the area of the triangle as A.
We are given that the base of the triangle is 10 feet long. So, we have b = 10.
The base angles A are decreasing at a rate of 2° per second. This means that dA/dt = -2°/s.
To find the rate of change of the area (dA/dt), we need to find an equation that expresses A (the area) in terms of b (the base).
The formula to calculate the area of a triangle is A = (1/2) * b * h, where b is the base and h is the height.
In an isosceles triangle, the height is related to the base by the formula h = (b/2) * tan(A/2).
Substituting the given values, we have A = (1/2) * 10 * (10/2) * tan(A/2).
Simplifying further, A = 25 * tan(A/2).
To find dA/dt, we differentiate both sides of this equation with respect to t.
dA/dt = d/dt (25 * tan(A/2)).
Using the chain rule, we have:
dA/dt = d/dA (25 * tan(A/2)) * dA/dt.
The derivative of tan(A/2) with respect to A/2 is sec^2(A/2) * (1/2).
So, dA/dt = 25 * sec^2(A/2) * (1/2) * dA/dt.
Plugging in the given value of dA/dt = -2°/s and the desired value of A = 45°, we can calculate the rate of change of the area:
dA/dt = 25 * sec^2(45/2) * (1/2) * (-2°/s).
To convert degrees to radians, we multiply by π/180:
dA/dt = 25 * sec^2(45/2) * (1/2) * (-2°/s) * (π/180).
Calculating this expression will give you the rate of change of the area of the isosceles triangle when the base angles are 45°.