A right triangle has a fixed base of length 6 meters and a height that is increasing at a rate of 2
meters/second. At what rate is the length of the hypotenuse increasing when the height is 8 meters?
If the hypotenuse has length z, then
h^2+36 = z^2
when h=8, z=10
2h dh/dt = 2z dz/dt
or,
h dh/dt = z dz/dt
So, when h=8,
8*2 = 10 dz/dt
dz/dt = 1.6 m/s
1/5
To solve this problem, we can use the Pythagorean theorem, which states that for any right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.
Let's denote the hypotenuse as h, the height as x, and the base as b.
According to the Pythagorean theorem, we have:
h^2 = x^2 + b^2
Taking the derivative of both sides with respect to time t, we get:
d(h^2)/dt = d(x^2)/dt + d(b^2)/dt
Differentiating both sides of the equation using chain rule, we get:
2h * dh/dt = 2x * dx/dt + 2b * db/dt
Since we are given that the base has a fixed length of 6 meters, we can substitute b = 6 in the equation:
2h * dh/dt = 2x * dx/dt + 2(6) * db/dt
Now we need to find the values of h, x, dx/dt, and db/dt when x = 8.
Given:
b = 6 meters (fixed)
x = 8 meters (height)
dx/dt = 2 meters/second (rate of change of height)
To find h, we can substitute b = 6 and x = 8 into the Pythagorean theorem equation:
h^2 = x^2 + b^2
h^2 = (8)^2 + (6)^2
h^2 = 64 + 36
h^2 = 100
h = 10 meters (hypotenuse)
Now we can substitute h = 10, x = 8, and dx/dt = 2 into the derivative equation:
2(10) * dh/dt = 2(8) * (2) + 2(6) * db/dt
Simplifying the equation:
20 * dh/dt = 16 + 12 * db/dt
We want to find dh/dt, so we rearrange the equation:
20 * dh/dt = 16 + 12 * db/dt
dh/dt = (16 + 12 * db/dt)/20
Since the base length is not changing, db/dt = 0. Therefore:
dh/dt = (16 + 12 * 0)/20
dh/dt = 16/20
dh/dt = 0.8 meters/second
Therefore, when the height is 8 meters, the length of the hypotenuse is increasing at a rate of 0.8 meters/second.
To find the rate at which the length of the hypotenuse is increasing, we can use the chain rule from calculus. The formula for the length of the hypotenuse in a right triangle is given by the Pythagorean theorem:
c = sqrt(a^2 + b^2)
where c is the length of the hypotenuse, and a and b are the lengths of the two legs of the triangle.
In this case, the base length is fixed at 6 meters, so we only need to find the rate at which the hypotenuse is changing with respect to the height. Let's define the variables:
a = 6 (fixed base length)
b = h (height)
c = hypotenuse length
To find dc/dt, the rate at which the hypotenuse is changing with respect to time, we need to find the derivative of c with respect to t (time) and use the chain rule:
dc/dt = (dc/da) * (da/dt) + (dc/db) * (db/dt)
Since da/dt = 0 (the base length is fixed), we can ignore the first term. Now, let's find the second term:
dc/db = (1/2) * (a^2 + b^2)^(-1/2) * (2b)
Before substituting the values, we need to find db/dt. Given that the height is increasing at a rate of 2 meters/second:
db/dt = 2
Now, substitute the values into the formula and simplify:
dc/db = (1/2) * (6^2 + h^2)^(-1/2) * (2h) = h / sqrt(36 + h^2)
Finally, substitute the current height (h = 8) into the formula to find the rate at which the hypotenuse is increasing:
dc/dt = (h / sqrt(36 + h^2)) * (db/dt) = (8 / sqrt(36 + 8^2)) * 2 = 16 / sqrt(100) = 16/10 = 1.6 meters/second
Therefore, when the height is 8 meters, the length of the hypotenuse is increasing at a rate of 1.6 meters/second.