"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.
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
Well, it seems like Superman is causing quite the mathematical dilemma! Let's see if we can solve this riddle using a bit of clown humor.
First, let's imagine that Superman is not just any superhero - he's Super Clown Man! He's flying in the sky above WHS High School, spreading laughter and joy to all. Now, we know that superheroes are known for their incredible speeds, but Super Clown Man likes to take things slow. He wants to make sure everyone gets a chance to enjoy his antics, after all.
So, 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, Super Clown Man decides to slow down a bit. He doesn't want to zoom past the school and miss out on all the fun!
With a mischievous grin, Super Clown Man decreases his flying speed to a leisurely 10 mph. As the distance between him and the school increases at a rate of 40 mph, he chips in an extra 30 mph to maintain the distance of exactly 10 miles. So, his overall flying speed would now be 40 mph!
Now, we know this might not be the most scientifically accurate answer, but hey, we're clowns here! Remember, laughter is the best solution to many problems. So, let's just enjoy the silly image of Super Clown Man soaring through the sky at a leisurely pace, spreading joy to all.
Keep smiling and don't forget to have fun with your math problems too!
To solve this problem, we can use the Pythagorean theorem and differentiate it with respect to time:
Let:
x = distance between Superman and WHS high school
y = height of Superman above the ground
According to the given information:
x = √(r^2 - y^2)
r = 10 (constant)
r' = 40 mph (rate at which x is increasing with respect to time)
y = 6 (constant)
Differentiating both sides of the equation with respect to time (t), you get:
dx/dt = (1/2)(r^2 - y^2)^(-1/2) * (2r * dr/dt - 2y * dy/dt)
Substituting the given values:
dx/dt = (1/2)(10^2 - 6^2)^(-1/2) * (2 * 10 * 40 - 2 * 6 * 0)
dx/dt = (1/2)(64)^(-1/2) * (800)
dx/dt = (1/2)(1/8) * (800)
dx/dt = (1/16) * (800)
dx/dt = 50 mph
Therefore, Superman is flying at 50 mph when the distance between him and WHS high school is exactly 10 miles, and the rate is increasing at 40 mph.
To solve this problem, we can use the Pythagorean Theorem and the chain rule to find the rate at which Superman is flying. Let's go through the steps:
1. Define the variables:
- Let x represent the horizontal distance between Superman and WHS High School.
- Let y represent the vertical distance between Superman and the ground.
- Let r represent the direct distance (hypotenuse) between Superman and WHS High School.
- Let r' represent the rate at which the distance between Superman and WHS High School is changing.
2. Apply the Pythagorean Theorem:
Based on the given information, we have:
x^2 + y^2 = r^2
3. Differentiate both sides of the Pythagorean equation with respect to time t:
Since r = 10, x = 8, y = 6, and r' = 40, we can differentiate the equation implicitly using the chain rule:
2x * (dx/dt) + 2y * (dy/dt) = 2r * (dr/dt)
Here, dx/dt represents the rate at which x is changing (horizontal speed of Superman), and dy/dt represents the rate at which y is changing (vertical speed of Superman). We are looking to find dx/dt.
4. Solve for dx/dt:
Substituting the given values into the derived equation and solving for dx/dt, we have:
2(8) * (dx/dt) + 2(6) * 0 = 2(10) * 40
16(dx/dt) = 800
dx/dt = 800/16
dx/dt = 50 mph
So, according to the calculations, Superman is flying at a speed of 50 mph horizontally (dx/dt) 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 (r').