Question 1

Find the volume of a cap of a sphere with radius r = 100 and height h = 39  &#10;

Accepted Answer

7857 &#960;

Question 2

Find the volume of a pyramid with height   and base an equilateral triangle with side a =    &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The volume of a pyramid is given by the formula $V=\frac{1}{3}Bh$ where $B$ is the area of the base and $h$ is the height. In this case, the pyramid has an equilateral triangle as a base with side length $a = 11ea8a3b_b6ff_eb50_9fff_233e3a059310_TB6897_11 11ea8a3b_b6ff_eb51_9fff_57600df81180_TB6897_00$.

To find the area of the base, we first need to find its height. Since the triangle is equilateral, we can use the Pythagorean theorem to find the height. Let $h_1$ be the height of one of the 30-60-90 triangles formed by the altitude to the base. Then, $h_1^2 = a^2 - (\frac{1}{2}a)^2 = \frac{3}{4}a^2$ and $h_1 = \frac{\sqrt{3}}{2}a$. Thus, the height of the equilateral triangle base is $h_b = 2h_1 = \sqrt{3}a$.

The area of an equilateral triangle with side length $a$ is given by the formula $A = \frac{\sqrt{3}}{4}a^2$. Thus, the area of the base is $B = \frac{\sqrt{3}}{4}a^2$.

Substituting the given values, we get:

$V = \frac{1}{3} \cdot \frac{\sqrt{3}}{4}a^2 \cdot 11ea8a3b_b6ff_eb4f_9fff_9dfd348754c0_TB6897_11$

$V = \frac{1}{3} \cdot \frac{\sqrt{3}}{4} \cdot (11ea8a3b_b6ff_eb50_9fff_233e3a059310_TB6897_11)^2 \cdot 11ea8a3b_b6ff_eb4f_9fff_9dfd348754c0_TB6897_11$

Simplifying and converting to hexadecimal (TB6897_11), we get:

$V = 11ea8a3b_b700_1263_9fff_bda832ac5241_TB6897_11$

Therefore, the answer is (B).

Question 3

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E) None

Accepted Answer

11ea8a3b_b6ff_4f00_9fff_69cc6a259c3e_TB6897_11

Question 4

Find the volume of the solid obtained by rotating the region bounded by   and   about the y-axis.&#10;

Accepted Answer

The answer of Find the volume of the solid obtained...

Question 5

Find the volume of the solid obtained by rotating the region bounded by   about the line  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Find the volume of the solid obtained...

Question 6

Find the volume of the frustum of a pyramid with square base of side   square top of side   and height    &#10;

Accepted Answer

The answer of Find the volume of the frustum of...

Question 7

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.  &#10;

Accepted Answer

The answer of Use a computer algebra system to find...

Question 8

The volume of a solid torus (the donut-shaped solid shown in the figure)with r =   and R =   is

Accepted Answer

The formula for the volume of a solid torus is V = pi^2 * R^2 * r, where R is the distance from the center of the torus to the center of the tube and r is the radius of the tube. Plugging in the given values, we get V = pi^2 * (11ea8a3b_b700_fcde_9fff_4ba59b53ae6b_TB6897_11)^2 * (11ea8a3b_b700_fcdd_9fff_477577226991_TB6897_11) = 11ea8a3b_b701_23ef_9fff_a1efeb402aca_TB6897_11, which matches the given answer.

Question 9

The base of S is the parabolic region   Cross-sections perpendicular to the y axis are squares. Find the volume of S.&#10;

Accepted Answer

The answer of The base of S is the parabolic...

Question 10

Find the volume common to two spheres,each with radius r =   if the center of each sphere lies on the surface of the other sphere.&#10;

Accepted Answer

The answer of Find the volume common to two spheres,each...

Question 11

Find the volume of the solid obtained by rotating the region bounded by   and   about the line  &#10;

Accepted Answer

The answer of Find the volume of the solid obtained...

Question 12

Find the volume of the solid obtained by rotating the region bounded by   about the x-axis.&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Find the volume of the solid obtained...

Question 13

The base of S is a circular region with boundary curve   Cross-sections perpendicular to the x axis are isosceles right triangles with hypotenuse in the base. Find the volume of S.&#10;

Accepted Answer

The answer of The base of S is a circular...

Deck 7: Extension B: Differential Equations