Answer:
The solution is obtained by adding the two equations.
The solution is: (x, y) = (
,
)
Step-by-step explanation:
We are given two equations with two variables. The strategy is to eliminate one variable and solve for both the variables.
The two equations are:


Adding both the equations, we get:



Substituting the value of 'x', we get the value of y.
We substitute in (2). [Can be substituted in any equation].
We get: y = 2x - 1



So, we get the corresponding values of x and y which is the solution of the two equations.
Answer:
15 seconds
Step-by-step explanation:
If you make a table of values for the dog and the squirrel using d = rt, then the rates are easy: the dog's rate is 150 and the squirrel's is 100. The t is what we are looking for, so that's our unknown, and the distance is a bit tricky, but let's look at what we know: the dog is 200 feet behind the squirrel, so when the dog catches up to the squirrel, he has run some distance d plus the 200 feet to catch up. Since we don't know what d is, we will just call it d! Now it seems as though we have 2 unknowns which is a problem. However, if we solve both equations (the one for the dog and the one for the squirrel) for t, we can set them equal to each other. Here's the dog's equation:
d = rt
d+200 = 150t
And the squirrel's:
d = 100t
If we solve both for t and set them equal to each other we have:

Now we can cross multiply to solve for d:
150d = 100d + 20,000 and
50d = 20,000
d = 400
But we're not looking for the distance the squirrel traveled before the dog caught it, we are looking for how long it took. So sub that d value back into one of the equations we have solved for t and do the math:

That's 1/4 of a minute which is 15 seconds.
Answer:
So, the volume is:

Step-by-step explanation:
We get the limits of integration:

We use the spherical coordinates and we calculate a triple integral:
![V=\int_0^{2\pi}\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\int_0^4 \rho^2 \sin \varphi \, d\rho\, d\varphi\, d\theta\\\\V=\int_0^{2\pi}\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \sin \varphi \left[\frac{\rho^3}{3}\right]_0^4\, d\varphi\, d\theta\\\\V=\int_0^{2\pi}\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \sin \varphi \cdot \frac{64}{3} \, d\varphi\, d\theta\\\\V=\frac{64}{3} \int_0^{2\pi} [-\cos \varphi]_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \, d\theta\\\\V=\frac{64}{3} \int_0^{2\pi} \sqrt{2} \, d\theta\\\\](https://tex.z-dn.net/?f=V%3D%5Cint_0%5E%7B2%5Cpi%7D%5Cint_%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5E%7B%5Cfrac%7B3%5Cpi%7D%7B4%7D%7D%5Cint_0%5E4%20%20%5Crho%5E2%20%5Csin%20%5Cvarphi%20%5C%2C%20d%5Crho%5C%2C%20d%5Cvarphi%5C%2C%20d%5Ctheta%5C%5C%5C%5CV%3D%5Cint_0%5E%7B2%5Cpi%7D%5Cint_%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5E%7B%5Cfrac%7B3%5Cpi%7D%7B4%7D%7D%20%5Csin%20%5Cvarphi%20%5Cleft%5B%5Cfrac%7B%5Crho%5E3%7D%7B3%7D%5Cright%5D_0%5E4%5C%2C%20d%5Cvarphi%5C%2C%20d%5Ctheta%5C%5C%5C%5CV%3D%5Cint_0%5E%7B2%5Cpi%7D%5Cint_%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5E%7B%5Cfrac%7B3%5Cpi%7D%7B4%7D%7D%20%5Csin%20%5Cvarphi%20%5Ccdot%20%5Cfrac%7B64%7D%7B3%7D%20%5C%2C%20d%5Cvarphi%5C%2C%20d%5Ctheta%5C%5C%5C%5CV%3D%5Cfrac%7B64%7D%7B3%7D%20%5Cint_0%5E%7B2%5Cpi%7D%20%5B-%5Ccos%20%5Cvarphi%5D_%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5E%7B%5Cfrac%7B3%5Cpi%7D%7B4%7D%7D%20%20%5C%2C%20d%5Ctheta%5C%5C%5C%5CV%3D%5Cfrac%7B64%7D%7B3%7D%20%5Cint_0%5E%7B2%5Cpi%7D%20%5Csqrt%7B2%7D%20%5C%2C%20d%5Ctheta%5C%5C%5C%5C)
we get:
![V=\frac{64}{3} \int_0^{2\pi} \sqrt{2} \, d\theta\\\\V=\frac{64\sqrt{2}}{3}\cdot[\theta]_0^{2\pi}\\\\V=\frac{128\sqrt{2}\pi}{3}](https://tex.z-dn.net/?f=V%3D%5Cfrac%7B64%7D%7B3%7D%20%5Cint_0%5E%7B2%5Cpi%7D%20%5Csqrt%7B2%7D%20%5C%2C%20d%5Ctheta%5C%5C%5C%5CV%3D%5Cfrac%7B64%5Csqrt%7B2%7D%7D%7B3%7D%5Ccdot%5B%5Ctheta%5D_0%5E%7B2%5Cpi%7D%5C%5C%5C%5CV%3D%5Cfrac%7B128%5Csqrt%7B2%7D%5Cpi%7D%7B3%7D)
So, the volume is:

Answer:
42+34+14=90
Step-by-step explanation:
42 people are male
34 people study biology
14 males study biology (thats both)