hectictar

\(\frac{\text{volume of the sphere}}{\text{surface area of the sphere}}=4\text{ cm} \\~\\ \frac{\frac43\pi r^3}{4\pi r^2}=4\text{ cm} \\~\\ \frac{\frac43\not{\pi} r\cdot\not{r^2}}{4\not{\pi}\not{r^2}}=4\text{ cm} \\~\\ \frac{\frac43r}{4}=4 \text{ cm} \\~\\ \frac43r=16\text{ cm} \\~\\ r=16/\frac43\text{ cm} \\~\\r=12\text{ cm}\)

false

midpoint of (-1, 4) and (5, 4) \(=(\frac{-1+5}{2},\frac{4+4}{2}) \\~\\ =(\frac{4}{2},\frac{8}{2}) \\~\\ =(2,4)\)

You can see it pretty clearly on a graph:

(sin a)² + (cos a)²= 1

(-4/5 )² + (cos a)² = 1

\({\cos a = \sqrt{1-(-\frac{4}{5})^2}=\sqrt{1-\frac{16}{25}}=\sqrt{\frac9{25}}=\frac35}\)
 

(sin b)² + (cos b)²= 1

(5/13)² + (cos b)² = 1

\({\cos b =\sqrt{1-(\frac{5}{13})^2} = \sqrt{1-\frac{25}{169}}=\sqrt{\frac{144}{169}}=\frac{12}{13} }\)

cos(a - b) = (cos a)(cos b) + (sin a)(sin b)

cos(a - b) = (3/5)(12/13)    + (-4/5)(5/13)

cos(a - b) = 36/65 - 4/13

cos(a - b) = 16/65

S \(= \text{the midpoint of PQ} \\~\\ = \text{the midpoint of (-7, 3) and (1, 3)} \\~\\ = ( \frac{-7+1}2,\frac{3+3}{2}) \\~\\ = (-3, 3) \\~\\\)

T \(= \text{the midpoint of QR} \\~\\ = \text{the midpoint of (1, 3) and (1, -3)} \\~\\ = ( \frac{1+1}{2},\frac{3+-3}{2}) \\~\\ = (1, 0) \\~\\\)

length of ST \(=\sqrt{(-3-1)^2+(3-0)^2} \\~\\ =\sqrt{(-4)^2+(3)^2} \\~\\ =\sqrt{16+9} \\~\\ =\sqrt{25} \\~\\ =5\)

You got it right!! 

Remember that θ is the angle that a line from the origin to the point makes with the positive x axis. That is, the line to the right of the y-axis.

To find r, we can use the Pythagorean theorem.

(-52)² + 75² = r²

\(\sqrt{2704+5625}\) = r

\(\sqrt{8329}\) = r

r ≈ 91.26

To find θ, we can take the arctan(75/-52) ...

arctan(75/-52) ≈ -55.27º

But..the calculator gives us an angle that lies in the fourth quadrant.

The tangent values repeat every 180º ...so just add 180º to get the angle that has the same tangent in the second quadrant.

-55.27º + 180º = 124.73º ≈ θ

So...the polar coordinates are (91.26, 124.73º)  

Maybe this can help:

Hahaha!! 

Ohh.....so did I pass?? 

I may not be able to explain any better...but I will try!

Here is just a graph of the point (21, 48).

Theta is the angle that a line from the origin to the point makes with the positive x-axis.

The second one is the same graph, just with some more markings on it.

Look at the second picture. This is just a right triangle! And...in every right triangle,

tan = opposide / adjacent

So..

tan θ = 48/21

θ = atan(48/21)

θ ≈ 66.371º

And we can use the Pythagorean theorem to find the hypotenuse...which is r.

21² + 48² = r²

r = \(\sqrt{441+2304}\approx52.393\) ...just as you found!