Studdy AI Logo
Math  /  Geometry

Question13. A street light is mounted at the top of a 15 -ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 5ft/s5 \mathrm{ft} / \mathrm{s} along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole?

Studdy Solution

STEP 1

1. The street light is at the top of a 15-ft-tall pole.
2. The man is 6 ft tall.
3. The man walks away from the pole at a speed of 5 ft/s5 \, \text{ft/s}.
4. We need to find how fast the tip of his shadow is moving when he is 40 ft from the pole.

STEP 2

1. Set up the relationship between the distances using similar triangles.
2. Differentiate the relationship with respect to time.
3. Substitute the known values and solve for the rate of change of the shadow's tip.

STEP 3

Set up the relationship using similar triangles:
Let x x be the distance of the man from the pole, and s s be the length of the shadow. The triangles formed by the pole and the shadow, and the man and the shadow, are similar. Therefore, the ratios of their corresponding sides are equal:
15x+s=6s\frac{15}{x + s} = \frac{6}{s}

STEP 4

Differentiate the relationship with respect to time t t :
First, solve the equation for s s :
15s=6(x+s)15s = 6(x + s)
15s=6x+6s15s = 6x + 6s
9s=6x9s = 6x
s=23xs = \frac{2}{3}x
Differentiate both sides with respect to t t :
dsdt=23dxdt\frac{ds}{dt} = \frac{2}{3} \frac{dx}{dt}

STEP 5

Substitute the known values and solve for dsdt\frac{ds}{dt}:
Given dxdt=5 ft/s\frac{dx}{dt} = 5 \, \text{ft/s}, substitute this into the differentiated equation:
dsdt=23×5\frac{ds}{dt} = \frac{2}{3} \times 5
dsdt=103 ft/s\frac{ds}{dt} = \frac{10}{3} \, \text{ft/s}
The rate at which the tip of the shadow is moving is the sum of the man's speed and the rate of change of the shadow's length:
d(x+s)dt=dxdt+dsdt=5+103\frac{d(x+s)}{dt} = \frac{dx}{dt} + \frac{ds}{dt} = 5 + \frac{10}{3}
d(x+s)dt=153+103=253 ft/s\frac{d(x+s)}{dt} = \frac{15}{3} + \frac{10}{3} = \frac{25}{3} \, \text{ft/s}
The tip of the shadow is moving at a rate of:
253 ft/s \boxed{\frac{25}{3} \, \text{ft/s}}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord