Question

Tangent Line Approximation Problem: If a right triangle has legs 6 and 8, its hypotenuse is 10. The triangle will be inscribed
within a circle with area 25pi . (The hypotenuse will be the diameter of the circle.)

A. Suppose one leg of the triangle is known to be exactly 6, but the other leg is known to
be 8 with an error of +/-h . What are x, f(x), and a in this problem?

Answer 1

B. Use a tangent line approximation to estimate the area of the circumscribed circle.
C. Now consider the sphere that just contains the triangle (so the hypotenuse is the
diameter of the sphere). Use a tangent line approximation to estimate the volume of
this sphere
This is free response; can someone help me get started? .

Answer 2

we get options?

Answer 3

I'm sorry, what do you mean?

Answer 4

at means it looks like your taking a test and wanting us to pick the right option :)

Answer 5

Oh! No, those are separate questions. This is a homework practice assignment for Tangent Line Approximation in Calc 1.

Answer 6

then im prolly butchering this thing up :)

Answer 7

That looks like my picture! But what does the text say?

Answer 8

not the center, im making note of that even tho it looks centery like

Answer 9

centernal?

Answer 10

I have no idea :P The diameter is 10 though, correct?

Answer 11

hyp is diameter, oy; its like reading a novel by steinbeck

Answer 12

what is an "a"? is there a pic that goes with this?

Answer 13

Oh! The equation for tangent line approximation is f(x) is approximately equal to f(a)+f'(a)(x-a)

Answer 14

If we were just trying to change the area of the triangle, I believe, because 8 is +/- h, 8 would be a and 8+h or 8-h would be x.

Answer 15

the diameter is the hyp; and the hyp depend on 8+h or 8-h

6^2 + (8-h)^2 = lesser margin ^2

6^2 + (8+h)^2 = higher margin ^2

Answer 16

Wait! The area of the circle is \[\pi r^2\] The r is dependent on \[6^2 + (8\pm h)^2\]

Answer 17

correct

Answer 18

So my function that I'll be using will be a synthesis of these two. Any suggestions of a function to represent this?

Answer 19

The area of the circle, I mean.

Answer 20

replace r^2 by its substitution

Answer 21

pi(36+(8+-h)^2)

Answer 22

Does the portion in parentheses need to be squared? It's pi r^2, right?

Answer 23

Oh. Nevermind.

Answer 24

you see it right :)

Answer 25

Yes! I do! Thanks so much for your help. Okay, moving onto the sphere portion...The same thing applies with the radius being substituted except it will be substituted into the volume of a sphere equation?

Answer 26

yes, since the volume of a sphere is dependant on "r"

4/3 pi r^3 is the volume

so make a neccesary adjustment for the extra "r"

r = sqrt(36 + (8+-h)^2 )

Answer 27

Thank you so much! I should be able to continue.

Answer 28

good luck :)