Question

g A person who is 6 feet tall is walking away from a lamp post at the rate of 40 feet per minute. When the person is 10 feet from the lamp post, his shadow is 20 feet long. Find the rate at which the length of the shadow is increasing when he is 30 feet from the lamp post.

Answer 1

Answer:

80 ft/min

Step-by-step explanation:

Let h represent the height of the person shadow, x represent the distance between the person and the lamppost, y represent the length of the man shadow.

Therefore:

$\frac{h}{x+y} =\frac{6}{y} \\\\substituting\ x=10,y=20:\\\\\frac{h}{10+20} =\frac{6}{20}\\\\\frac{h}{30} =\frac{6}{20}\\\\h=30*6/20\\\\h=9\ feet\\\\Therefore:\\\frac{h}{x+y} =\frac{6}{y}\\\\\frac{9}{x+y} =\frac{6}{y}\\\\9y=6x+6y\\\\9y-6y=6x\\\\3y=6x\\\\y=2x\\\\The\ person\ is\ walking\ from\ the\ lamppost\ at\ 40ft/min(\frac{dx}{dt}=40\ ft/min) \\\\y=2x\\\\Differentiating\ with\ respect\ to\ t:\\\\\frac{dy}{dt} =\frac{d}{dt}(2x)\\\\\frac{dy}{dt} =2\frac{dx}{dt}$

$\frac{dy}{dt} =2(40\ ft/min)\\\\\frac{dy}{dt}=80\ ft/min$

Answer 2

The rate at which the length of the shadow is increasing when he is 30 feet away from the lamp post is  80 $\frac{ft}{min}$.

The height of the person shadow is to be = h

Given ,

The distance between the person and the man = x = 10

The length of the man shadow is = y = 20

By the similar triangle property corresponding sides of similar triangles are in the same ratio.

$\frac{h}{x+y} = \frac{6}{y}$

$\frac{h}{10+20} = \frac{6}{20}$

20h = 60 + 120

20h = 180

h = $\frac{180}{20}$

h = 9feet

Therefore,

$\frac{h}{x+y} = \frac{6}{y} \\\frac{9}{x+y} = \frac{6}{y}$

9y= 6x + 6y

9y-6y = 6x

6x = 3y

y = $\frac{6}{3}x$

y = 2x

Differentiating both the sides with respect t,

$\frac{dy}{dt} = 2\frac{dx}{dt}$

we know that ;

$\frac{dx}{dt} = 40 \frac{ft}{sec}$

Therefore, by substitute the value

$\frac{dy}{dt} = 2 . 40\frac{ft}{min}$

$\frac{dy}{dt}$ = 80$\frac{ft}{min}$.

The rate at which the length of the shadow is increasing is 80$\frac{ft}{min}$  .

For more details about the Rate change problem click the link given below.

https://brainly.com/question/20414862