1. ## help

1. a certain tree measures 102.6 feet in distance(the circumference).show how to find the distance straight through the center of the tree(the diameter) without measuring it directly.then find this value to the nearest tenth of a foot.
2. suppose a rope wrapped around earth at the equator(earth's circuference C).then think of adding d feet to the rope's lenght so it can now circle earth at a distance h feet above the equator at all points.
a. write an equation to model this situation.
b. solve d.
c. how much rope do you need to insert if you want the rope to circle earth 500 feet above the equator?

2. Hello, zasi!

1. A certain tree measures 102.6 feet in circumference.
Show how to find the distance straight through the center of the tree(the diameter)
without measuring it directly, then find this value to the nearest tenth of a foot.

You're expected to know this formula: .$\displaystyle C \:=\:\pi d$
. . where $\displaystyle D$ is the diameter of the circle and $\displaystyle C$ is its circumference.

We are given: .$\displaystyle C = 102.6$
. . Substitute into the formula: .$\displaystyle 102.6\:=\:\pi D\quad\Rightarrow\quad D \:=\:\frac{102.6}{\pi}$

Therefore: .$\displaystyle D \:=\:32.65859432\:\approx\:32.7$ feet.

2. Suppose a rope wrapped around earth at the Equator.
Then think of adding $\displaystyle d$ feet to the rope's length
so it can now circle earth at a distance $\displaystyle h$ feet above the Equator at all points.

a. Write an equation to model this situation.
b. Solve for $\displaystyle d$.
c. How much rope do you need to insert if you want the rope
. . to circle earth 500 feet above the Equator?

Circumference formula: .$\displaystyle C \:=\:2\pi R$
. . where $\displaystyle R$ = radius of the circle.

Let $\displaystyle R$ = radius of the earth (in feet).

Then the original rope has length $\displaystyle C_1 \:=\:2\pi R$ feet.

The new rope has length $\displaystyle C_2\:=\:2\pi(R + h)$ .
I hope you see why.

The difference of the two circumferences is $\displaystyle d$.
. . $\displaystyle C_2 - C_1\:=\:d\quad\Rightarrow\quad 2\pi(R + h) - 2\pi R \:=\:d$ .(a)

(b) .$\displaystyle d \:=\:2\pi h$

(c) .We want $\displaystyle h = 500$ feet.
. . . Therefore: .$\displaystyle d \:=\:2\pi(500) \:=\:3141.592654\:\approx\:3141.6$ feet.

3. Originally Posted by zasi
1. a certain tree measures 102.6 feet in distance(the circumference).show how to find the distance straight through the center of the tree(the diameter) without measuring it directly.then find this value to the nearest tenth of a foot.
2. suppose a rope wrapped around earth at the equator(earth's circuference C).then think of adding d feet to the rope's lenght so it can now circle earth at a distance h feet above the equator at all points.
a. write an equation to model this situation.
b. solve d.
c. how much rope do you need to insert if you want the rope to circle earth 500 feet above the equator?
Hello,

you are supposed to know:

$\displaystyle p = 2 \cdot \pi \cdot r$. p stands for perimeter.

to 1.

Plug in the value you know and solve for r. Then calculate d = 2r:

102.6'=2*π*r
r=(102.6)/(2*π) ≈ 16.329'. Therefore d = 2r = 32.6'

to 2.

Use the following equations:

$\displaystyle C=2 \cdot \pi \cdot R$
$\displaystyle C+d=2 \cdot \pi \cdot (R+h)$

b) Solve for d:

$\displaystyle C+d=2 \cdot \pi \cdot (R+h) \Longleftrightarrow d= 2 \cdot \pi \cdot (R+h) - C=2 \cdot \pi \cdot R+ 2 \cdot \pi \cdot h - 2 \cdot \pi \cdot R$

$\displaystyle d=2 \cdot \pi \cdot h$

c) Now h = 500'. Plug in this value and solve for d:

$\displaystyle d=2 \cdot \pi \cdot 500' \Longleftrightarrow d = 1000 \cdot \pi \approx 3141.6'$

EB