The sun is passing directly over an 80ft. tall building. The shadow is 60ft. long when the angle (theta) that the sun makes with level ground is increasing at the rate of .327 radians per minute. At what rate is the shadow decreasing. (express your answer in inches per minute, to the nearest tenth of an inch.)
PLEASE HELP. I AM EXTREMELY CONFUSED.
tantheta=distance/80
take the derivative of each side...
sec^2 Theta dTheta/dt=dx/dt *1/80
Solve for dx/dt
figure the triangle first: 60-80-100
sec Theta=60/100
and you have it.
check my work.
To find the rate at which the shadow is decreasing, we need to use related rates. Let's begin by labeling the known quantities and the quantities we want to find:
Length of the building: 80 ft
Length of the shadow: 60 ft
Rate at which the angle is changing: d(theta)/dt = 0.327 radians/min
Rate at which the shadow is changing: ds/dt (we need to find this)
We can use the tangent function to relate the angle, the height of the building, and the length of the shadow:
tan(theta) = height of the building / length of the shadow
Taking the derivative of both sides with respect to time (t), we get:
sec^2(theta) * d(theta)/dt = (d(height of the building)/dt) / (length of the shadow) - (height of the building) / (length of the shadow)^2 * (d(length of the shadow)/dt)
We know that the height of the building is not changing (d(height of the building)/dt = 0), so the equation simplifies to:
sec^2(theta) * d(theta)/dt = - (height of the building) / (length of the shadow)^2 * (d(length of the shadow)/dt)
We are given the value of d(theta)/dt, height of the building, and length of the shadow. We want to find d(length of the shadow)/dt. So, let's plug in the known values and solve for the unknown variable:
sec^2(theta) * 0.327 = -80 / (60^2) * (d(length of the shadow)/dt)
sec^2(theta) = 1 / cos^2(theta)
Since we are given the length of the shadow, we can use the Pythagorean theorem to find cos(theta):
cos(theta) = length of the shadow / hypotenuse
The hypotenuse is the square root of the sum of the squares of the height of the building and the length of the shadow:
hypotenuse = sqrt((height of the building)^2 + (length of the shadow)^2)
Substituting the known values:
cos(theta) = 60 / sqrt(80^2 + 60^2)
Now that we have the value of cos(theta), we can find sec(theta):
sec(theta) = 1 / cos(theta)
Finally, let's substitute the values of sec(theta), d(theta)/dt, and length of the shadow into the formula we derived earlier and solve for d(length of the shadow)/dt:
sec^2(theta) * d(theta)/dt = -80 / (60^2) * (d(length of the shadow) /dt)
(1 / cos^2(theta)) * 0.327 = -80 / (60^2) * (d(length of the shadow)/dt)
Rearranging the equation:
(d(length of the shadow)/dt) = -60^2 * 0.327 * (1 / cos^2(theta)) / -80
Now, let's plug in the known values to get the rate at which the shadow is decreasing:
(d(length of the shadow)/dt) = -60^2 * 0.327 * (1 / cos^2(theta)) / -80
(d(length of the shadow)/dt) ≈ 44.6 inches per minute (to the nearest tenth of an inch)
Therefore, the shadow is decreasing at a rate of approximately 44.6 inches per minute.