A right triangle has legs of 24 inches and 32 inches whose sides are changing. The short leg is increasing by 8 in/sec and the long leg is shrinking at 5 in/sec. What is the rate of change of the hypotenuse?
Oh, triangles, they're always changing sides! Anyway, let's get down to business. We have a right triangle with legs of 24 inches and 32 inches, and they're changing. The short leg is increasing by 8 in/sec, while the long leg is playing it cool and shrinking at 5 in/sec. Now, we want to know the rate of change of the hypotenuse. Well, let me do a quick calculation while juggling some numbers here. *throws numbers in the air* *catches them* Ah, I got it! The rate of change of the hypotenuse is approximately 10.124 in/sec. Now, that's some speedy business going on there, I hope they all can keep up with the changes!
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 (c) is equal to the sum of the squares of the other two sides (a and b).
Given that the legs of the right triangle are changing, we need to find their rates of change first.
Let's call the short leg a, the long leg b, and the hypotenuse c. We are given:
da/dt = 8 in/sec (rate of change of the short leg)
db/dt = -5 in/sec (rate of change of the long leg)
Now, let's differentiate both sides of the Pythagorean theorem with respect to time (t):
(c^2) = (a^2) + (b^2)
Differentiating both sides with respect to time (t):
2c(dc/dt) = 2a(da/dt) + 2b(db/dt)
Simplifying:
2c(dc/dt) = 2a(8) + 2b(-5)
c(dc/dt) = 8a - 10b
Now, substitute the given values for a and b:
c(dc/dt) = 8(24) - 10(32)
c(dc/dt) = 192 - 320
c(dc/dt) = -128 in^2/sec
Therefore, the rate of change of the hypotenuse is -128 in^2/sec.
To find the rate of change of the hypotenuse, we need to 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 short leg as x, the long leg as y, and the hypotenuse as z. We are given that the short leg is increasing at a rate of 8 in/sec and the long leg is shrinking at a rate of 5 in/sec.
Using the Pythagorean theorem, we have x^2 + y^2 = z^2.
Differentiating both sides of the equation with respect to time t, we get:
2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt).
Since we are looking for the rate of change of the hypotenuse (dz/dt), we need to solve for it.
Plugging in the given values, we have:
2(24)(8) + 2(32)(-5) = 2z(dz/dt).
Simplifying the equation, we get:
384 - 320 = 2z(dz/dt).
64 = 2z(dz/dt).
Dividing both sides of the equation by 2z, we get:
dz/dt = 64/(2z).
To find dz/dt, we need to know the value of z. However, we don't have enough information to determine its value. Therefore, the rate of change of the hypotenuse (dz/dt) cannot be determined with the given information.
H^2 = S^2 + L^2
this is a 3-4-5 triangle ... H = 40
2 H dH/dt = 2 S dS/dt + 2 L dL/dt
40 dH/dt = (24 * 8) - (32 * 5)