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A 6 foot man walks away from a light sitting atop a pole 16 feet above ground. (a) How long is his shadow when he is 8 feet from the pole? (b) What is the distance from the base of the pole to the tip of his shadow?

(a) $4.8 \mathrm{ft}$(b) $12.8 \mathrm{ft}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 9

Elements of Geometry

Derivatives

Campbell University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

02:02

A person who is 6 feet tal…

03:06

Length of a Shadow $\mathr…

01:57

Length of a Shadow A man i…

00:56

A man is walking away from…

Moving Shadow A man 6 ft t…

04:19

A street light is mounted …

00:40

A 6 -foot man is standing …

this word problem is painting a picture for us. So any time you're given descriptions, describe something concrete. It's always a good idea to start with a picture if you can. This does not have to be a great picture, but a picture nonetheless. Okay, so I have a light. Okay, so there's my light pole and I have a man walking away from the light pole and they're both casting a shadow. So there's the shadow. There's our man. Okay, what do I know? Let's put in numbers. Well, I have a 6 ft tall man. So the height of the man is six and the light is sitting on top of a 16 ft pole. So those are two pieces. I know those are the and then I know that he's 8 ft from the pole, so I know that piece as well. What I want to find is the length of his shadow. So I'm going to put that as an excuse. I definitely want to find X. I also want to know the distance from the base of the pole to the tip of his shadow, which is going to be eight plus X. So those are the two pieces I want his shadow and the entire distance from the shadow to the lamppost. So how will we go about setting this up? But what we have here are similar triangles. The triangle that the lamppost makes with the tip of the shadow. Way out here is the large triangle, and the man to the tip of the shadow is the smaller triangle. Similar triangles. I can put the sides together in a similar proportion. So let's do Let's do big to small. That's going to how we're going to set this up. So the big triangle is 16. The small triangle is six. I know those that pair of sides. I know that ratio 16 to 6 now. The base, the Big triangle is eight plus X, and the small triangle is X. I'll just say it's very It's a common mistake to say, Oh, the big triangles. Eight. I see two numbers there. I want one to be 11 to be the other. It's not the case. The big triangle is the whole base after the eight plus the X. Don't forget that plus X there. Okay, now we can simplify this before we do our cross multiplication. Let's make these numbers as small as possible. I can divide the six by two and the 16 by two to make them just a little smaller. If I cross multiply, I get eight. X equals 24 plus three x and I'm going to solve for X. So five X equals 24 or X equals 24 5th. So the length of his shadow is 20 far fifth feet. If you want to put this in decimals, you could say that this is 4.8 ft. Either one would work. Okay, so how about that entire length? Well, I just need to have X plus eight common denominator means instead of an eight, I'm really going to be adding 40/5. And that's going to give me 64 5th feet as the distance from the base of the lamp post of the tip of his shadow. Again, if you want to do this in decimals, you could write This is 12.8 ft. So those are the lengths that we were trying to find

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