The base of an isosceles triangle is 8 feet long. If the altitude is 6 feet long and is increasing 3 inches per minute, are what rate are the base angles changing?
Handsome
To find the rate at which the base angles of the isosceles triangle are changing, we need to use the given information about the rate of change of the altitude and the relationship between the base and the altitude.
Let's start by labeling the isosceles triangle. The base of the triangle is given as 8 feet, and the altitude is given as 6 feet. Let's call the two equal sides of the triangle "a" and the base angles "θ".
Next, we need to find the relationship between the base and the altitude of an isosceles triangle. In an isosceles triangle, the altitude is perpendicular to the base and bisects it. This means that the altitude divides the base into two equal segments.
So, in our triangle, each segment of the base is 8/2 = 4 feet long.
Now, let's determine how the base angles change when the altitude is increasing. Since the altitude is increasing at a rate of 3 inches per minute, we need to convert this to feet per minute to match the units of the base and the segments.
1 foot = 12 inches, so 3 inches = 3/12 = 1/4 feet.
Therefore, the altitude is increasing at a rate of 1/4 feet per minute.
Since the two segments of the base are equal in length, we can use the concept of similar triangles to find the rate of change of the base angles.
We have the relationship: (change in altitude) / (change in base) = (change in opposite side) / (change in adjacent side).
Let's represent the rate of change of the base angles as dθ/dt (the derivative of θ with respect to t).
Applying the concept of similar triangles:
(dθ/dt) / (1/4) = (dθ/dt) / 4 = (d(a/2)/dt) / 4.
Since we know that (d(a/2)/dt) = (d(base)/dt) = 0 (because the base length is constant at 8 feet), we can simplify the equation:
(dθ/dt) / 4 = 0.
Now, we can solve for dθ/dt:
dθ/dt = 0 * 4 = 0.
Therefore, the rate at which the base angles are changing is zero.
Thank you
we have
tanθ = h/4
sec^2θ dθ/dt = 1/4 dh/dt
Now just plug in your numbers.