The volume of a sphere is given by:

So, we need to deduct this equation. We will walk through Calculus on the concept of a solid of revolution that is a solid figure that is obtained by rotating a plane curve around some straight line (the axis of revolution<span>) that lies on the same plane. We know from calculus that:
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![V=\pi \int_{a}^{b}[f(x)]^{2}dx](https://tex.z-dn.net/?f=V%3D%5Cpi%20%5Cint_%7Ba%7D%5E%7Bb%7D%5Bf%28x%29%5D%5E%7B2%7Ddx)
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Then, according to the concept of solid of revolution we are going to rotate a circumference shown in the figure, then:
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Isolationg y:
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So,
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![V=\pi \int_{a}^{b}[\sqrt{r^{2}-x^{2}}]^{2}dx](https://tex.z-dn.net/?f=V%3D%5Cpi%20%5Cint_%7Ba%7D%5E%7Bb%7D%5B%5Csqrt%7Br%5E%7B2%7D-x%5E%7B2%7D%7D%5D%5E%7B2%7Ddx)
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being -r and r the limits of this integral.
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Solving:
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![V=\pi[r^{2}x-\frac{x^{3}}{3}]\right|_{-r}^{r}](https://tex.z-dn.net/?f=V%3D%5Cpi%5Br%5E%7B2%7Dx-%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D%5D%5Cright%7C_%7B-r%7D%5E%7Br%7D)
Finally:
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m<ABD = 52 Given
BC bisects <ABD Given
m<ABC = m<CBD Definition of angle bisector.
m<ABC +m<CBD = m<ABD Angle addition postulate.
m<ABC +m<CBD = 52 Substitution
m<ABC + m<ABC= 52 Substitution
2m<ABC = 52 Combining like angles
m<ABC = 26 Division property of equality.
Answer:
Step-by-step explanation:
To verify, a number needs to be substituted for x in both expressions. Use order of operations to simplify and find the value. The value needs to be the same for both expressions to prove
There are no equations to choose from but it should look like the following.
a= # of adult tickets
s= # of student tickets
QUANTITY EQUATION
a + s= 560
COST EQUATION
$8a + $3s= $2905
***If you have to also solve for the number of adults and students, here are the steps.
STEP 1:
multiply quantity equation by -8
-8(a + s)= -8(560)
-8a - 8s= -4480
STEP 2:
add cost equation and step 1 equation
$8a + $3s= $2905
-8a - 8s= -4480
a term cancels out to zero
-5s= -1575
divide both sides by -5
s= 315 students
STEP 3:
substitute s=315 in quantity equation
a + s= 560
a + 315= 560
subtract 315 from both sides
a= 245 adults
ANSWER:
Quantity Equation
a + s= 560
Cost Equation
$8a + $3s= $2905
Hope this helps! :)