The center of the circle is (3,9) so the answer is A
ANSWER
![y = 2x -4](https://tex.z-dn.net/?f=y%20%3D%202x%20-4)
EXPLANATION
Part a)
Eliminating the parameter:
The parametric equation is
![x = 5 + ln(t)](https://tex.z-dn.net/?f=x%20%3D%205%20%2B%20%20ln%28t%29%20)
![y = {t}^{2} + 5](https://tex.z-dn.net/?f=y%20%3D%20%20%7Bt%7D%5E%7B2%7D%20%20%2B%205)
From the first equation we make t the subject to get;
![x - 5 = ln(t)](https://tex.z-dn.net/?f=x%20-%205%20%3D%20%20ln%28t%29%20)
![t = {e}^{x - 5}](https://tex.z-dn.net/?f=t%20%3D%20%20%7Be%7D%5E%7Bx%20-%205%7D%20)
We put it into the second equation.
![y = { ({e}^{x - 5}) }^{2} + 5](https://tex.z-dn.net/?f=y%20%3D%20%20%7B%20%28%7Be%7D%5E%7Bx%20-%205%7D%29%20%7D%5E%7B2%7D%20%20%2B%205)
![y = { ({e}^{2(x - 5)}) } + 5](https://tex.z-dn.net/?f=y%20%3D%20%20%7B%20%28%7Be%7D%5E%7B2%28x%20-%205%29%7D%29%20%7D%20%20%2B%205)
We differentiate to get;
![\frac{dy}{dx} = 2 {e}^{2(x - 5)}](https://tex.z-dn.net/?f=%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%20%3D%202%20%7Be%7D%5E%7B2%28x%20-%205%29%7D%20)
At x=5,
![\frac{dy}{dx} = 2 {e}^{2(5 - 5)}](https://tex.z-dn.net/?f=%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%20%3D%202%20%7Be%7D%5E%7B2%285%20-%205%29%7D%20)
![\frac{dy}{dx} = 2 {e}^{0} = 2](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%20%20%3D%202%20%7Be%7D%5E%7B0%7D%20%20%3D%202)
The slope of the tangent is 2.
The equation of the tangent through
(5,6) is given by
![y-y_1=m(x-x_1)](https://tex.z-dn.net/?f=y-y_1%3Dm%28x-x_1%29)
![y - 6 = 2(x - 5)](https://tex.z-dn.net/?f=y%20-%206%20%3D%202%28x%20-%205%29)
![y = 2x - 10 + 6](https://tex.z-dn.net/?f=y%20%3D%202x%20-%2010%20%2B%206)
![y = 2x -4](https://tex.z-dn.net/?f=y%20%3D%202x%20-4)
Without eliminating the parameter,
![\frac{dy}{dx} = \frac{ \frac{dy}{dt} }{ \frac{dx}{dt} }](https://tex.z-dn.net/?f=%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%20%3D%20%20%5Cfrac%7B%20%5Cfrac%7Bdy%7D%7Bdt%7D%20%7D%7B%20%5Cfrac%7Bdx%7D%7Bdt%7D%20%7D%20)
![\frac{dy}{dx} = \frac{ 2t}{ \frac{1}{t} }](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%20%20%3D%20%20%5Cfrac%7B%202t%7D%7B%20%20%5Cfrac%7B1%7D%7Bt%7D%20%7D%20)
![\frac{dy}{dx} = 2 {t}^{2}](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%20%20%3D%20%202%20%7Bt%7D%5E%7B2%7D%20)
At x=5,
![5 = 5 + ln(t)](https://tex.z-dn.net/?f=5%20%3D%205%20%2B%20%20ln%28t%29%20)
![ln(t) = 0](https://tex.z-dn.net/?f=%20ln%28t%29%20%20%3D%200)
![t = {e}^{0} = 1](https://tex.z-dn.net/?f=t%20%3D%20%20%7Be%7D%5E%7B0%7D%20%20%3D%201)
This implies that,
![\frac{dy}{dx} = 2 {(1)}^{2} = 2](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%20%20%3D%20%202%20%7B%281%29%7D%5E%7B2%7D%20%20%3D%202)
The slope of the tangent is 2.
The equation of the tangent through
(5,6) is given by
![y-y_1=m(x-x_1)](https://tex.z-dn.net/?f=y-y_1%3Dm%28x-x_1%29)
![y - 6 = 2(x - 5) =](https://tex.z-dn.net/?f=y%20-%206%20%3D%202%28x%20-%205%29%20%3D)
![y = 2x -4](https://tex.z-dn.net/?f=y%20%3D%202x%20-4)
Answer:
y=-(4x/3) -1
Step-by-step explanation:
Slope=m= -4/3
y intercept = c = -1
now using equation;
y= mx+c
y= (-4/3)x+(-1)
y=-(4x/3) -1
Answer:
V =5292 in ^3
Step-by-step explanation:
To find the volume for a rectangular prism
V = l*w*h
V = 6 * 36 * 24.5
V =5292 in ^3