Hi! a lot of jobs require math in the real world. for instance, a architect has to get all the math correct on a building, otherwise one wrong measurement and it could fall. a lot of jobs involve math.
hope this helps!:)
Okay here's an example so if we have 2, 3/2*1/2 we convert the mixed numbers to improper fractions. So in order to solve a mixed number you add the denominator, numerator and whole to get 7/2*1.2
So Multiple
7/2*1/2
Refine the fractions
7/2*2
Multiple the numbers 2*2=4 to get
7/4
and that equals 1, 3/4
We have some couples of angles that are equal
Answer: A. ![A'(t)=\frac{4356\pi}{(t+11)^{3}}-\frac{527076\pi}{(t+11)^{5}}](https://tex.z-dn.net/?f=A%27%28t%29%3D%5Cfrac%7B4356%5Cpi%7D%7B%28t%2B11%29%5E%7B3%7D%7D-%5Cfrac%7B527076%5Cpi%7D%7B%28t%2B11%29%5E%7B5%7D%7D)
B. A'(5) = 1.76 cm/s
Step-by-step explanation: <u>Rate</u> <u>of</u> <u>change</u> measures the slope of a curve at a certain instant, therefore, rate is the derivative.
A. Area of a circle is given by
![A=\pi.r^{2}](https://tex.z-dn.net/?f=A%3D%5Cpi.r%5E%7B2%7D)
So to find the rate of the area:
![\frac{dA}{dt}=\frac{dA}{dr}.\frac{dr}{dt}](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7Bdt%7D%3D%5Cfrac%7BdA%7D%7Bdr%7D.%5Cfrac%7Bdr%7D%7Bdt%7D)
![\frac{dA}{dr} =2.\pi.r](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7Bdr%7D%20%3D2.%5Cpi.r)
Using ![r(t)=3-\frac{363}{(t+11)^{2}}](https://tex.z-dn.net/?f=r%28t%29%3D3-%5Cfrac%7B363%7D%7B%28t%2B11%29%5E%7B2%7D%7D)
![\frac{dr}{dt}=\frac{726}{(t+11)^{3}}](https://tex.z-dn.net/?f=%5Cfrac%7Bdr%7D%7Bdt%7D%3D%5Cfrac%7B726%7D%7B%28t%2B11%29%5E%7B3%7D%7D)
Then
![\frac{dA}{dt}=2.\pi.r.[\frac{726}{(t+11)^{3}}]](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7Bdt%7D%3D2.%5Cpi.r.%5B%5Cfrac%7B726%7D%7B%28t%2B11%29%5E%7B3%7D%7D%5D)
![\frac{dA}{dt}=2.\pi.[3-\frac{363}{(t+11)^{2}}].\frac{726}{(t+11)^{3}}](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7Bdt%7D%3D2.%5Cpi.%5B3-%5Cfrac%7B363%7D%7B%28t%2B11%29%5E%7B2%7D%7D%5D.%5Cfrac%7B726%7D%7B%28t%2B11%29%5E%7B3%7D%7D)
Multipying and simplifying:
![\frac{dA}{dt}=\frac{4356\pi}{(t+11)^{3}} -\frac{527076\pi}{(t+11)^{5}}](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7Bdt%7D%3D%5Cfrac%7B4356%5Cpi%7D%7B%28t%2B11%29%5E%7B3%7D%7D%20-%5Cfrac%7B527076%5Cpi%7D%7B%28t%2B11%29%5E%7B5%7D%7D)
The rate at which the area is increasing is given by expression
.
B. At t = 5, rate is:
![A'(5)=\frac{4356\pi}{(5+11)^{3}} -\frac{527076\pi}{(5+11)^{5}}](https://tex.z-dn.net/?f=A%27%285%29%3D%5Cfrac%7B4356%5Cpi%7D%7B%285%2B11%29%5E%7B3%7D%7D%20-%5Cfrac%7B527076%5Cpi%7D%7B%285%2B11%29%5E%7B5%7D%7D)
![A'(5)=\frac{4356\pi}{4096} -\frac{527076\pi}{1048576}](https://tex.z-dn.net/?f=A%27%285%29%3D%5Cfrac%7B4356%5Cpi%7D%7B4096%7D%20-%5Cfrac%7B527076%5Cpi%7D%7B1048576%7D)
![A'(5)=\frac{2408693760\pi}{4294967296}](https://tex.z-dn.net/?f=A%27%285%29%3D%5Cfrac%7B2408693760%5Cpi%7D%7B4294967296%7D)
![A'(5)=1.76268](https://tex.z-dn.net/?f=A%27%285%29%3D1.76268)
At 5 seconds, the area is expanded at a rate of 1.76 cm/s.