One leg of a right angle triangle is always 6 meters long, and the other
leg is increasing at a rate of 2 meters per second. Find the rate of change of the hypotenuse when it is 10 meters long.
plz help
h^2 = 6^2 + x^2
h dh/dt = x dx/dt
when h=10, x=8, so just plug in the numbers
10 dh/dt = 8*2
dh/dt = 8/5 m/s
Well, well, well, it seems like this triangle is getting quite a leg workout! Let's get to the math circus and solve this problem.
We're given that one leg of the triangle is always 6 meters long, and the other leg is increasing at a rate of 2 meters per second. Our main attraction is to find the rate of change of the hypotenuse when it is 10 meters long.
To solve this, we'll use a good old Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's call the increasing leg length x, and the hypotenuse h. According to the theorem, we have the equation:
x^2 + 6^2 = h^2
Now, let's take the derivative of both sides with respect to time:
2x*(dx/dt) = 2h*(dh/dt)
Since we know that x is increasing at a rate of 2 meters per second, we can substitute dx/dt with 2.
Plugging in the relevant values, we have:
2x*2 = 10*(dh/dt)
Simplifying this equation, we get:
4x = 10*(dh/dt)
So, when x = 10 and using the equation above, we can find the rate of change of the hypotenuse, dh/dt.
dh/dt = (4x) / 10
dh/dt = (4 * 10) / 10
dh/dt = 4
Voila! The rate of change of the hypotenuse when it is 10 meters long is 4 meters per second. Hope I could add some amusement to this mathematical trapeze act!
To find the rate of change of the hypotenuse when it is 10 meters long, we can use the Pythagorean theorem. According to the theorem, the square of the hypotenuse is equal to the sum of the squares of the other two legs.
Given that one leg is always 6 meters long, let's call the other leg "x". So, the length of the hypotenuse is given by the equation:
h^2 = 6^2 + x^2
Differentiating both sides of the equation with respect to time (t), we get:
2h * dh/dt = 0 + 2x * dx/dt
Since we are interested in finding the rate of change of the hypotenuse (dh/dt), we can substitute the known values:
h = 10 meters
dx/dt = 2 meters per second (as given in the question)
x = ? (to be determined)
Plugging these values into the equation, we have:
2 * 10 * dh/dt = 0 + 2x * 2
Simplifying:
20 * dh/dt = 4x
Since we want to find the rate of change of the hypotenuse, we need to determine the value of x when the hypotenuse is 10 meters long. Using the Pythagorean theorem, we can solve for x:
10^2 = 6^2 + x^2
100 = 36 + x^2
x^2 = 64
x = 8 meters
Substituting the value of x into the equation, we have:
20 * dh/dt = 4 * 8
Simplifying further:
20 * dh/dt = 32
Finally, we can calculate the rate of change of the hypotenuse (dh/dt) by dividing both sides of the equation by 20:
dh/dt = 32/20
dh/dt = 1.6 meters per second
Therefore, the rate of change of the hypotenuse when it is 10 meters long is 1.6 meters per second.
To find the rate of change of the hypotenuse, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's denote the length of the increasing leg as x (in meters) at a given time t (in seconds). We know that one leg is always 6 meters long. So, in the right triangle, the legs are 6 meters and x meters (where x = 6 + 2t).
We can use the Pythagorean theorem to express the length of the hypotenuse (h) in terms of the other sides:
h^2 = 6^2 + x^2
Now, since x = 6 + 2t, we can substitute it into the equation:
h^2 = 6^2 + (6 + 2t)^2
Simplifying the equation:
h^2 = 36 + 36 + 24t + 4t^2
h^2 = 4t^2 + 24t + 72
To find the rate of change of the hypotenuse (dh/dt) when it is 10 meters long, we need to differentiate both sides of the equation with respect to time (t):
2h * (dh/dt) = 8t + 24
Now we have an equation in terms of the hypotenuse h, and we know that h = 10 meters. Substituting the value of h, we can solve for the rate of change of the hypotenuse:
2(10) * (dh/dt) = 8t + 24
20 * (dh/dt) = 8t + 24
Now, we can find the rate of change (dh/dt) by dividing both sides by 20:
(dh/dt) = (8t + 24) / 20
Simplifying the equation:
(dh/dt) = (2t + 6) / 5
Therefore, the rate of change of the hypotenuse is (2t + 6) / 5 meters per second.