Answer:
The rate of change of radius is 0.0530 ft/sec.
Step-by-step explanation:
Given : Helium is pumped into a spherical balloon at a rate of 3 cubic feet per second.
To find : How fast is the radius increasing after 2 minutes?
Solution :
The volume of the sphere is ![V=\frac{4}{3}\pi r^3](https://tex.z-dn.net/?f=V%3D%5Cfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3)
Differentiating w.r.t. r,
![\frac{dV}{dr}=4\pi r^2](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdr%7D%3D4%5Cpi%20r%5E2)
We have given, Helium is pumped into a spherical balloon at a rate of 3 cubic feet per second.
i.e. ![\frac{dV}{dt}=3](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdt%7D%3D3)
Integrate w.r.t. t,
![V=3t+C](https://tex.z-dn.net/?f=V%3D3t%2BC)
Assuming V=0 when t=0 then C=0
So, ![\frac{4}{3}\pi r^3=3t](https://tex.z-dn.net/?f=%5Cfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3%3D3t)
![r^3=\frac{t}{4\pi}](https://tex.z-dn.net/?f=r%5E3%3D%5Cfrac%7Bt%7D%7B4%5Cpi%7D)
![r=\sqrt[3]{\frac{t}{4\pi}}](https://tex.z-dn.net/?f=r%3D%5Csqrt%5B3%5D%7B%5Cfrac%7Bt%7D%7B4%5Cpi%7D%7D)
Now, we apply chain rule
i.e. ![\frac{dV}{dt}=\frac{dV}{dr}\cdot \frac{dr}{dt}](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdt%7D%3D%5Cfrac%7BdV%7D%7Bdr%7D%5Ccdot%20%5Cfrac%7Bdr%7D%7Bdt%7D)
![3=4\pi(\frac{t}{4\pi})^{\frac{2}{3}}\cdot \frac{dr}{dt}](https://tex.z-dn.net/?f=3%3D4%5Cpi%28%5Cfrac%7Bt%7D%7B4%5Cpi%7D%29%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%5Ccdot%20%5Cfrac%7Bdr%7D%7Bdt%7D)
The radius increasing after 2 minutes i.e. at ![t=2\times 60=120](https://tex.z-dn.net/?f=t%3D2%5Ctimes%2060%3D120)
![3=4\pi(\frac{120}{4\pi})^{\frac{2}{3}})\cdot \frac{dr}{dt}](https://tex.z-dn.net/?f=3%3D4%5Cpi%28%5Cfrac%7B120%7D%7B4%5Cpi%7D%29%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%29%5Ccdot%20%5Cfrac%7Bdr%7D%7Bdt%7D)
![3=56.56\cdot \frac{dr}{dt}](https://tex.z-dn.net/?f=3%3D56.56%5Ccdot%20%5Cfrac%7Bdr%7D%7Bdt%7D)
![\frac{dr}{dt}=\frac{3}{56.56}](https://tex.z-dn.net/?f=%5Cfrac%7Bdr%7D%7Bdt%7D%3D%5Cfrac%7B3%7D%7B56.56%7D)
![\frac{dr}{dt}=0.0530](https://tex.z-dn.net/?f=%5Cfrac%7Bdr%7D%7Bdt%7D%3D0.0530)
Therefore, the rate of change of radius is 0.0530 ft/sec.
Answer:
<h2>31.25%.</h2>
Step-by-step explanation:
The experimental probability is the ratio between the number of times an event occurs and the total number of trials.
In this case, the total number of trials is 64, because Page roles it 64 times.
Notice that the probability is about one event or the other, that indicates we need to sum both events.
![P=\frac{14+6}{64} =\frac{20}{64} \approx 0.3125](https://tex.z-dn.net/?f=P%3D%5Cfrac%7B14%2B6%7D%7B64%7D%20%3D%5Cfrac%7B20%7D%7B64%7D%20%5Capprox%200.3125)
Multiplying by 100, we have a probability of 31.25%.
Therefore, the experimental probability of getting three or four is 31.25%.
Equation given:
15x - 6y = 48
First, turn 15x into -15x and plug it into 48.
15x - 6y = 48
-15x -15x
The equation should look like this:
-6y = -15x + 48
Divide -6y to 48 and -15x.
y =
- 8
Answer:
Option B.(1,8)
Step-by-step explanation:
Let
x ----> the number of hours worked
y ---->the amount earned in dollars
we know that
To find out the unit rate divide the total amount earned by the total hours worked
so
![\frac{\$48}{6}=\$8\ per\ hour](https://tex.z-dn.net/?f=%5Cfrac%7B%5C%2448%7D%7B6%7D%3D%5C%248%5C%20per%5C%20hour)
That means
For x=1 hour, y=$8
therefore
The point that represent the unit rate is (1,8)
2.8 miles.
My Work/Reasoning:
First we find the rate. 7/60 mpm (miles per minutes)
Then we multiply that by 24. 7/60 * 24
Finally, we solve the equation and learn that Chang would run 2.8 miles in 24 minutes.