the lengths of two legs of a right triangle depend on time. One, whose length is x increase at the rate of 5 feet per second, while the other, of length y, decreases at the rate of 6 feet per second. At what rate is the hypotenuse changing when x=3 and y=4? is the hypotenuse increasing or decreasing then?
given: dx/dt = 5 , dy/dt = -6
when x=3 and y = 4 , hypotenuse = 5 , (the infamous 3-4-5 right-angled triangle)
H^2 = x^2 + y^2
2H dH/dt = 2x dx/dt + 2y dy/dt
dH/dt = (x dx/dt + y dy/dt)/H
= (3(5) + 4(-6))/5
= -9/5
at that moment of time, the hypotenuse is decreasing at a rate of 9/5 ft/s
To find the rate at which the hypotenuse is changing, 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:
c^2 = a^2 + b^2,
where c is the length of the hypotenuse, and a and b are the lengths of the legs.
Differentiating both sides with respect to time t, we have:
(d/dt) (c^2) = (d/dt) (a^2 + b^2).
Using the chain rule, we get:
2c * (dc/dt) = 2a * (da/dt) + 2b * (db/dt).
Since we want to find the rate at which the hypotenuse is changing when x = 3 and y = 4, we can substitute these values into the equation.
Given:
da/dt = 5 ft/s (rate of increase of x),
db/dt = -6 ft/s (rate of decrease of y),
a = x = 3 (length of one leg),
b = y = 4 (length of the other leg).
Substituting these values into the equation, we have:
2c * (dc/dt) = 2(3) * (5) + 2(4) * (-6).
Simplifying the equation, we get:
2c * (dc/dt) = 30 - 48.
2c * (dc/dt) = -18.
Dividing both sides by 2c, we get:
(dc/dt) = -9/c.
Since we are interested in the rate at which the hypotenuse is changing, we need to find dc/dt when x = 3 and y = 4.
To find the length of the hypotenuse when x = 3 and y = 4, we can use the Pythagorean theorem:
c^2 = 3^2 + 4^2 = 9 + 16 = 25.
Taking the square root of both sides, we have:
c = 5.
Substituting c = 5 into the equation (dc/dt = -9/c), we get:
(dc/dt) = -9/5.
Therefore, when x = 3 and y = 4, the hypotenuse is changing at a rate of -9/5 ft/s. The negative sign indicates that the hypotenuse is decreasing.
To find the rate at which the hypotenuse is changing, we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides of the right triangle:
hypotenuse^2 = leg1^2 + leg2^2
Differentiating both sides of the equation with respect to time (t) using implicit differentiation, we get:
2 * hypotenuse * (rate of change of hypotenuse) = 2 * leg1 * (rate of change of leg1) + 2 * leg2 * (rate of change of leg2)
Now let's substitute the given information into the equation:
rate of change of hypotenuse = (leg1 * rate of change of leg1 + leg2 * rate of change of leg2) / hypotenuse
Given:
leg1, x = 3 ft (increasing at a rate of 5 ft/s)
leg2, y = 4 ft (decreasing at a rate of 6 ft/s)
rate of change of hypotenuse = (x * rate of change of x + y * rate of change of y) / hypotenuse
rate of change of hypotenuse = (3 * 5 + 4 * (-6)) / hypotenuse
Now, we need to find the value of the hypotenuse when x = 3 and y = 4. Using the Pythagorean theorem again, we can calculate it:
hypotenuse^2 = leg1^2 + leg2^2
hypotenuse^2 = 3^2 + 4^2
hypotenuse^2 = 9 + 16
hypotenuse^2 = 25
hypotenuse = 5 ft
So, let's substitute the values into the equation:
rate of change of hypotenuse = (3 * 5 + 4 * (-6)) / 5
rate of change of hypotenuse = (15 - 24) / 5
rate of change of hypotenuse = -9 / 5
Therefore, the rate at which the hypotenuse is changing when x = 3 and y = 4 is -9/5 ft/s. The negative sign indicates that the hypotenuse is decreasing.