"superman is in level flight 6 miles above ground. his flight plan takes him directly over whs high school. how fast is he flying when the distance between him and the whs high school is exactly 10 miles and the rate is increasing 40mph?"
i keep getting:
r = 10
r'=40
y=6
x=8
x'=50
but that is impossible since the derivative of x is still a leg of a right triangle and that it cannot be increasing faster than the hypotenuse. please help me i cannot find the answer.
reiny said:
Let his horizontal distance over the high school be x
let the distance between him and the school be y
I see a right-angled triangle and
x^2 + 6^2 = y^2
2x dx/dt + 0 = 2y dy/dt , #1
when x = 10, y^2 = 100 + 36, y = √136
in equation #1
2(10) dx/dt = 2(√136) (40)
dx/dt = 40√136/10
= 4√4 √ 34
= 8√34 mph or appr 46.65 mph
but if y = distance between him and school then y should = 10 and 8 should = 8, i am confused by the answer you gave me.
typo: 8 should = 8
i meant x should = 8
You are right, I subbed 10 into the wrong variable
Good catch.
when y = 10
x^2+ 6^2 = 10^2
x^2 =100-36= 64
x = √64 = 8
in my #1 from above post:
2(8) dx/dt = 2(10)(40)
dx/dt = 50 mph
I apologize for any confusion caused. Let's go through the problem step by step to determine the correct answer.
Given:
Superman is in level flight 6 miles above the ground.
The distance between Superman and WHS High School is exactly 10 miles.
The rate at which the distance is increasing is 40 mph.
Let's use the following variables:
x - Horizontal distance from Superman to WHS High School
y - Distance between Superman and WHS High School
We have a right-angled triangle formed by Superman, WHS High School, and the ground. Using the Pythagorean theorem, we can relate the variables x, y, and 6 (the vertical distance).
x^2 + 6^2 = y^2
x^2 + 36 = y^2
To determine the rate at which Superman is flying, we need to find dx/dt (the rate of change of x with respect to time) when y = 10 and dy/dt (the rate of change of y with respect to time) is 40 mph.
Differentiate the equation with respect to time (t):
2x (dx/dt) + 0 = 2y (dy/dt)
2x (dx/dt) = 2y (dy/dt)
x (dx/dt) = y (dy/dt)
dx/dt = y (dy/dt) / x
Plugging in the given values:
y = 10
dy/dt = 40
We need to find the value of x to calculate dx/dt.
Using the Pythagorean theorem equation, substituting y = 10:
x^2 + 36 = 100
x^2 = 64
x = ±8
Since we're dealing with distances, we take x = 8 miles.
Now, substitute y = 10, x = 8, and dy/dt = 40 into the equation:
dx/dt = y (dy/dt) / x
dx/dt = 10 (40) / 8
dx/dt = 400 / 8
dx/dt = 50 mph
Therefore, when the distance between Superman and WHS High School is exactly 10 miles and the rate of change of that distance is increasing at 40 mph, Superman is flying at a speed of 50 mph horizontally.
I hope this clarifies the steps and provides you with the correct answer. Let me know if you have any further questions!
I understand your confusion. Let me explain the process step by step.
First, we have the equation x^2 + 6^2 = y^2, which represents a right-angled triangle. x represents the horizontal distance over the high school, y represents the distance between Superman and the high school, and 6 represents the height of Superman above the ground.
Now, we differentiate the equation with respect to time (t), using the chain rule. This gives us:
2x(dx/dt) + 0 = 2y(dy/dt)
Since we are given that dy/dt = 40 mph and we want to find dx/dt when y = 10 miles, we can plug in these values:
2x(dx/dt) + 0 = 2(10)(40)
Simplifying this equation, we get:
2x(dx/dt) = 400
Now, we need to find the value of x when y = 10 miles. Plugging this into the original equation, we get:
x^2 + 6^2 = 10^2
x^2 + 36 = 100
x^2 = 64
x = 8
Substituting x = 8 into the equation we derived earlier, we have:
2(8)(dx/dt) = 400
16(dx/dt) = 400
dx/dt = 400/16
dx/dt = 25 mph
Therefore, Superman is flying at a speed of 25 mph when the distance between him and WHS high school is exactly 10 miles, and the rate at which the distance is increasing is 40 mph.
I hope this clarifies the solution for you.