90 degrees is 1/4 of the circle ( 90 * 4 = 360 degrees = full circle).
Divide the area of the circle by 4:
314 / 4 = 78.5 square inches is shaded.
Total = Principal * (1 + rate/n)^n*years where "n" is the number of compounding periods
Total = 9,250 * (1 + .07/52)^52*5
Total = <span>9,250 * (1.0013461538</span>)^260
<span><span>Total = 9,250 * 1.418733571
</span>
</span><span>Total = 13,123.29</span>
Source: Compound Interest Formulas
http://www.1728.org/compint3.htm
Answer:
![P[T \geq x]=e^{-\frac{1}{3}x}](https://tex.z-dn.net/?f=P%5BT%20%5Cgeq%20x%5D%3De%5E%7B-%5Cfrac%7B1%7D%7B3%7Dx%7D)
For x = 3 then
![P[T \geq 3]=0.368](https://tex.z-dn.net/?f=P%5BT%20%5Cgeq%203%5D%3D0.368)
Step-by-step explanation:
First we must consider the <em>“memoryless”</em> or <em>“forgetfulness”</em> property of the exponential distribution, which means that the likelihood of something happening in the future has no relation with past events.
Taking the above statement into account we have that the probability that T lasts more than a+b hours given that an event has lasted at least “a” hours equals the probability that T lasts more than “b” hours:
![P[T \geq a+b | T \geq a]=P[T \geq b]](https://tex.z-dn.net/?f=P%5BT%20%5Cgeq%20a%2Bb%20%7C%20T%20%5Cgeq%20a%5D%3DP%5BT%20%5Cgeq%20b%5D)
For our case we have:
![P[T \geq x+7 | T \geq 7]=P[T \geq x]](https://tex.z-dn.net/?f=P%5BT%20%5Cgeq%20x%2B7%20%7C%20T%20%5Cgeq%207%5D%3DP%5BT%20%5Cgeq%20x%5D)
To make calculations easier we resolve using the Exponential cumulative distribution, therefore:
![P[T \geq x]=1-P[T < x]](https://tex.z-dn.net/?f=P%5BT%20%5Cgeq%20x%5D%3D1-P%5BT%20%3C%20x%5D)
![=1 - (1 - e^{-\frac{1}{3}x})](https://tex.z-dn.net/?f=%3D1%20-%20%281%20-%20e%5E%7B-%5Cfrac%7B1%7D%7B3%7Dx%7D%29)
![=1 - 1 + e^{-\frac{1}{3}x}}](https://tex.z-dn.net/?f=%3D1%20-%201%20%2B%20e%5E%7B-%5Cfrac%7B1%7D%7B3%7Dx%7D%7D)
![=e^{-\frac{1}{3}x}](https://tex.z-dn.net/?f=%3De%5E%7B-%5Cfrac%7B1%7D%7B3%7Dx%7D)
Now, if x = 3 then,
![P[T > 3+7 | T> 7]=P[T \geq 3]=e^{-\frac{1}{3}3 }=e^{-1}=0.368](https://tex.z-dn.net/?f=P%5BT%20%3E%203%2B7%20%7C%20T%3E%207%5D%3DP%5BT%20%5Cgeq%203%5D%3De%5E%7B-%5Cfrac%7B1%7D%7B3%7D3%20%7D%3De%5E%7B-1%7D%3D0.368)
So, we can say that the probability that our friend waits at least 3 hours for the alarm to ring is 36.8%
The limit from the left is equal to the value of the function:
.. f(0) = 0
The limit from the right is the value that x^2 approaches as x approaches 0. That is 0^2 = 0.
Left and right limits are the same, so the limit as x approaches zero is 0.