A person 6 feet tall walks at a rate of 150 feet per minute toward a light tower whose searchlight is located 40 feet above the ground. Find the rate that the length of the shadow is changing.
Draw a diagram. If the man is x feet from the tower, and his shadow is s ft long, then using similar triangles,
s/6 = (x+s)/40
s = 3/17 x
ds/dt = 3/17 dx/dt
...
To find the rate at which the length of the shadow is changing, we can use similar triangles and apply related rates.
Let's denote the height of the person as "h", the distance between the person and the light tower as "x", and the length of the shadow as "y". We are given the following values:
- Height of the person (h) = 6 ft
- Rate at which the person is walking (dh/dt) = 150 ft/min
- Height of the light tower (40 ft)
We need to find the rate at which the length of the shadow (dy/dt) is changing.
Step 1: Set up a proportion using similar triangles. The ratio of the height of the person to the length of the shadow should be equal to the ratio of the distance from the person to the light tower to the length of the shadow.
h/y = x/y
Step 2: Differentiate both sides of the equation with respect to time (t).
(dh/dt)/y = (dx/dt)/y
Step 3: Substitute the given values and solve for (dx/dt), which represents the rate at which the distance between the person and the light tower is changing.
(150 ft/min)/y = (dx/dt)/y
Step 4: Cross-multiply and solve for (dx/dt).
150 ft/min = (dx/dt)
Therefore, the rate at which the distance between the person and the light tower is changing is 150 ft/min.
Step 5: Finally, substitute the value of (dx/dt) into the original proportion to find the rate at which the length of the shadow is changing (dy/dt).
(h/y) = (dx/dt)/y
(6 ft)/y = (150 ft/min)/y
Both sides of the equation have a factor of y, so we can cancel it out.
6 ft = 150 ft/min
Now, divide both sides of the equation by 6 ft to solve for (dy/dt).
(dy/dt) = (150 ft/min) / (6 ft)
(dy/dt) = 25 ft/min
Therefore, the rate at which the length of the shadow is changing is 25 ft/min.
To find the rate at which the length of the shadow is changing, we can use similar triangles. Let's assume that the person's height is represented by 'H' and the length of the shadow is represented by 'S.'
Using the concept of similar triangles, we can set up the following ratio:
H / S = (H + 40) / S'
Where 'S'' represents the rate at which the length of the shadow is changing. We need to find 'S''.
Now, we can solve for 'S'' by cross-multiplication:
(H)(S') = (H + 40)(S)
Rewriting the equation:
(H)(S') = (H)(S) + (40)(S)
We know that the person's height is 6 feet, and they walk at a rate of 150 feet per minute. Substituting these values into the equation:
6(S') = 6(S) + 40(S)
Simplifying the equation:
6S' = 6S + 40S
Combining like terms:
6S' = 46S
Now, we can solve for S' by dividing both sides of the equation by 6:
S' = 46S / 6
Simplifying further:
S' = 23/3 * S
So, the rate at which the length of the shadow is changing (S') is equal to 23/3 times the rate at which the person is walking (S).