A tight rope is stretched 30 feet above the ground between the Jay and the Tee buildings, which are 50 feet apart. A tightrope walker, walking at a constant rate of 2 feet per second from point A to point B, is illuminated by a spotlight 70 feet above point A, as shown in the diagram.
a. How fast is the shadow of the tightrope walker’s feet moving along the ground when she is midway between the buildings? (Indicate units of measure).
b. How far from point A is the tightrope walker when the shadow of her feet reaches the base of the Tee building? (Indicate unit of measure.)
c. How fast is the shadow of the tightrope walker’s feet moving up the wall of the Tee Building when she is 10 feet from point B? (Indicate units of measure.)
x/140=y/200
To solve these problems, we can use similar triangles and the concept of related rates. Let's go through each question step-by-step:
a. To find the speed at which the shadow of the tightrope walker's feet is moving along the ground when she is midway between the buildings, we need to find the rate at which the length of the shadow is changing with respect to time.
Let's call the distance from the tightrope walker to point A x feet. Since she is walking at a constant rate of 2 feet per second, the rate of change of x with respect to time is dx/dt = 2 ft/s.
We can create a similar triangle between the tightrope, the shadow, and the ground. The height of the similar triangle is the height of the tightrope, which is 30 feet, and the base of the similar triangle is the distance between the buildings, which is 50 feet.
Since the tightrope walker is midway between the buildings, the distance from the tightrope walker to the base of the triangle (the shadow) is half of the distance between the buildings, which is 50/2 = 25 feet.
Now, using the concept of similar triangles, we can set up the following proportion:
(height of the tightrope) / (length of the shadow) = (height of the similar triangle) / (length of the base of the similar triangle)
30 ft / length of the shadow = 30 ft / (50 ft)
Simplifying the equation, we have:
length of the shadow = (30 ft * 50 ft) / 30 ft = 50 ft
Now, we can differentiate the equation with respect to time to find the rate at which the length of the shadow is changing with respect to time:
d(length of the shadow) / dt = (d(50 ft)) / dt = 0 ft/s
Therefore, the speed at which the shadow of the tightrope walker's feet is moving along the ground when she is midway between the buildings is 0 ft/s.
b. To find the distance from point A to the tightrope walker when the shadow of her feet reaches the base of the Tee building, we need to find the length of the shadow at that point.
Since the tightrope walker is now at the base of the Tee building, the shadow's length will be equal to the distance between the buildings, which is 50 ft.
Now, we can use the similar triangles concept again to find the distance from point A to the tightrope walker.
Let's call this distance y. We can set up the following proportion:
(height of the tightrope) / (length of the shadow) = (distance between point A and the tightrope walker) / (distance between point A and the base of the Tee building)
30 ft / 50 ft = y / (50 ft)
Simplifying the equation, we have:
y = (30 ft * 50 ft) / 50 ft = 30 ft
Therefore, the tightrope walker is 30 feet from point A when the shadow of her feet reaches the base of the Tee building.
c. To find the speed at which the shadow of the tightrope walker's feet is moving up the wall of the Tee building when she is 10 feet from point B, we need to find the rate at which the height of the shadow is changing with respect to time.
Let's call the distance from the tightrope walker to point B z feet. Since she is walking at a constant rate of 2 feet per second, the rate of change of z with respect to time is dz/dt = -2 ft/s (negative because the distance is decreasing).
Using the similar triangles concept again, we can set up the following proportion:
(height of the tightrope) / (length of the shadow) = (height of the similar triangle) / (distance from the tightrope walker to point B)
30 ft / length of the shadow = 30 ft / z
Simplifying the equation, we have:
length of the shadow = (30 ft * z) / 30 ft = z ft
Now, we can differentiate the equation with respect to time to find the rate at which the length of the shadow is changing with respect to time:
d(length of the shadow) / dt = dz / dt = -2 ft/s
Therefore, the speed at which the shadow of the tightrope walker's feet is moving up the wall of the Tee building when she is 10 feet from point B is -2 ft/s.
To solve these problems, we can use similar triangles and the concepts of rates and proportions. Let's go through each question step by step:
a. To find the speed at which the shadow of the tightrope walker's feet is moving along the ground when she is midway between the buildings, we need to find the rate of change of the shadow's position with respect to time. Since the tightrope walker is walking at a constant rate of 2 feet per second, we can use this information.
Let's consider the midpoint between the buildings. At this point, the distance between the tightrope walker and the Jay building is 25 feet (half the distance of 50 feet). Using similar triangles, we can set up the following proportion:
(Height of the spotlight - Height of the shadow) / (Distance from the Jay building to the walker) = Height of the spotlight / Distance from the ground to the walker's shadow
(70 - 30) / 25 = 70 / x
Simplifying the equation, we get:
40 / 25 = 70 / x
Cross-multiplying, we have:
40x = 25 * 70
40x = 1750
Dividing both sides by 40, we find:
x = 43.75
Therefore, the distance from the ground to the tightrope walker's shadow when she is midway between the buildings is approximately 43.75 feet.
b. To find the distance from point A to the tightrope walker when the shadow of her feet reaches the base of the Tee building, we need to consider the similar triangles formed again. At this point, the distance between the tightrope walker and the Tee building is 50 feet. Let's set up the proportion:
(Height of the spotlight - Height of the shadow) / (Distance from point A to the walker) = Height of the spotlight / Distance from the ground to the walker's shadow
(70 - 30) / (50 - x) = 70 / x
Simplifying the equation, we get:
40 / (50 - x) = 70 / x
Cross-multiplying, we have:
40x = 70(50 - x)
40x = 3500 - 70x
110x = 3500
Dividing both sides by 110, we find:
x = 31.82
Therefore, the distance from point A to the tightrope walker when the shadow of her feet reaches the base of the Tee building is approximately 31.82 feet.
c. To find the speed at which the shadow of the tightrope walker's feet is moving up the wall of the Tee building when she is 10 feet from point B, we can again use similar triangles. At this point, the distance between the tightrope walker and the Tee building is 10 feet.
Using the same proportion as in part b:
(Height of the spotlight - Height of the shadow) / (Distance from point A to the walker) = Height of the spotlight / Distance from the ground to the walker's shadow
(70 - 30) / (50 - x) = 70 / x
Simplifying the equation, we get:
40 / (50 - x) = 70 / x
Cross-multiplying, we have:
40x = 70(50 - x)
40x = 3500 - 70x
110x = 3500
Dividing both sides by 110, we find:
x ≈ 31.82
Therefore, when the tightrope walker is 10 feet from point B, the speed at which the shadow of her feet is moving up the wall of the Tee building is approximately 31.82 feet per second.