Question 1

Thr area of the top side of a piece of sheet metal is 9 square feet. The sheet metal is submerged horizontally in 8 feet of water. Find the fluid force on the top side. Round your answer to One decimal place.
A)4,492.8 lb
B)4,420.8 lb
C)4,348.8 lb
D)4,622.4 lb
E)4,838.4 lb

Accepted Answer

The fluid force on a submerged surface can be calculated using the formula: Force = ρghA, where ρ (rho) is the density of the fluid (in this case, water, which is approximately 62.4 lb/ft³), g is the acceleration due to gravity (which is generally factored into the density for these types of problems), h is the depth of the fluid above the surface (8 feet), and A is the area of the surface (9 square feet). Therefore, the fluid force = 62.4 lb/ft³ * 8 ft * 9 ft² = 4,492.8 lb.

Question 2

Use the Theorem of Pappus to find the volume of the solid formed by revolving the region bounded by the graphs of $y = x , y = 10$, and $x = 0$ about the $x$-axis. Round your answer to two decimal places.
A) $1809.56$
B) $2787.64$
C) $2094.40$
D) $3141.59$
E) $3619.11$

Accepted Answer

The Theorem of Pappus states that the volume of a solid of revolution generated by revolving a plane figure about an external axis is equal to the product of the area of the figure and the distance traveled by its centroid. The region described is a right triangle with base 10 (along the y-axis from y=0 to y=10) and height 10 (along the x-axis from x=0 to x=10). The area of this triangle is $A = \frac{1}{2} \times base \times height = \frac{1}{2} \times 10 \times 10 = 50$. The centroid of a right triangle is located at one-third the base and one-third the height from the right angle, so the centroid is $\frac{10}{3}$ units from the x-axis. The distance traveled by the centroid when the region is revolved around the x-axis is $2\pi \times \frac{10}{3}$. Therefore, the volume is $V = A \times 2\pi \times \frac{10}{3} = 50 \times 2\pi \times \frac{10}{3} = \frac{1000\pi}{3} \approx 2094.40$.

Question 3

A circular plate of radius r feet is submerged vertically in a tank of fluid that weighs $w$ pounds per cubic foot. The center of the circle is $k ( k > r )$ feet below the surface of the fluid. The fluid force on the surface of the plate is given by $F = w k \left( \pi r ^ { 2 } \right)$ Find the fluid force on the circular plate as shown in the figure given $a = 5$ feet and $b = 2$ feet. Round your answer to one decimal place.

A) $5,489.0 \mathrm { lbs }$
B) $34,306.2 \mathrm { lbs }$
C) $6,861.2 \mathrm { lbs }$
D) $784.1 \mathrm { lbs }$
E) $2,352.4 \mathrm { lbs }$

Accepted Answer

The answer of A circular plate of radius r feet...

Question 4

$\text { Find } M _ { y } \text { for the lamina of uniform density } \rho \text { bounded by the graphs of the }$ equations $x = 4 y + 12$ and $x = y ^ { 2 }$.
A) $M _ { y } = \frac { 17,408 \rho } { 15 }$
B) $M _ { y } = \frac { 5,806 \rho } { 5 }$
C) $M _ { y } = \frac { 34,816 \rho } { 15 }$
D) $M _ { y } = \frac { 8,704 \rho } { 15 }$
E) $M _ { y } = \frac { 17,413 \rho } { 30 }$

Accepted Answer

To find $M_y$, the moment about the y-axis, for the lamina, we integrate the product of the density, the distance from the y-axis (x in this case), and the differential area element. The area is bounded by $x = 4y + 12$ and $x = y^2$, so we first find the intersection points by setting $4y + 12 = y^2$, which simplifies to $y^2 - 4y - 12 = 0$. Solving this quadratic equation gives the y-values of the intersection points. Then, we set up the integral of $x \rho \, dy$ across the y-boundaries, with $x$ expressed as the difference between the two bounding functions. The correct calculation leads to $M_y = \frac{17,408 \rho}{15}$, indicating that choice A is correct.

Question 5

The figure is the vertical side of a form for poured concrete that weighs 140.7 pounds per cubic foot. Dimensions in the figure are in feet. Determine force on this part of the concrete form. 
A) $2,251.20 \mathrm { lb }$
B) $46.90 \mathrm { lb }$
C) $234.50 \mathrm { lb }$
D) $562.80 \mathrm { lb }$
E) $3,376.80 \mathrm { lb }$

Accepted Answer

The answer of The figure is the vertical side of...

Question 6

A cylindrical gasoline tank is placed so that the axis of the cylinder is horizontal. Find the fluid force on a circular end of the tank if the tank is half full, assuming that the diameter is Feet and the gasoline weighs 42 pounds per cubic foot. 
A) $1,276 \mathrm { lb }$
B) $20,412 \mathrm { lb }$
C) $5,103 \mathrm { lb }$
D) $2,552 \mathrm { lb }$
E) $10,206 \mathrm { lb }$

Accepted Answer

The answer of A cylindrical gasoline tank is placed so...

Question 7

Find the fluid force of a triangular vertical plate submerged in water, where the dimensions are given in meters $( a = 15 , b = 6 , c = 4 )$ and the weight-density of water is 9800 newtons per cubic meter.

A) $6,174,000$ newtons
B) $4,851,000$ newtons
C) $8,379,000$ newtons
D) $4,483,500$ newtons
E) $14,994,000$ newtons

Accepted Answer

The answer of Find the fluid force of a triangular...

Question 8

Find the fluid force of a square vertical plate submerged in water, where meters and the weight-density of water is 9800 newtons per cubic meter. 
A) $171,225,600$ newtons
B) $376,476,800$ newtons
C) $40,336,800$ newtons
D) $349,585,600$ newtons
E) $25,930,800$ newtons

Accepted Answer

The answer of Find the fluid force of a square...

Question 9

Find the buoyant force of a rectangular solid of the given dimensions submerged in water so that the top side is parallel to the surface of the water. The buoyant force is the difference
Between the fluid forces on the top and bottom sides of the solid. Round your answer to two decimal Places. 
A)561.60 lb
B)748.80 lb
C)2,246.40 lb
D)249.60 lb
E)187.20 lb

Accepted Answer

The answer of Find the buoyant force of a rectangular...

Question 10

The figure is the vertical side of a form for poured concrete that weighs 140.7 pounds per cubic foot. Dimensions in the figure are in feet. Determine force on this part of the concrete form 
A) $6,753.60 \mathrm { lb }$
B) $703.50 \mathrm { lb }$
C) $10,130.40 \mathrm { lb }$
D) $1,688.40 \mathrm { lb }$
E) $140.70 \mathrm { lb }$

Accepted Answer

The answer of The figure is the vertical side of...

Question 11

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot. 
A) $93.6 \mathrm { lb }$
B) $3369.6 \mathrm { lb }$
C) $280.8 \mathrm { lb }$
D) $561.6 \mathrm { lb }$
E) $1684.8 \mathrm { lb }$

Accepted Answer

The answer of Find the fluid force on the vertical...

Question 12

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot. 
A) $140.4 \mathrm { lb }$
B) $561.6 \mathrm { lb }$
C) $280.8 \mathrm { lb }$
D) $1,123.2 \mathrm { lb }$
E) $2,246.4 \mathrm { lb }$

Accepted Answer

The answer of Find the fluid force on the vertical...

Question 13

A rectangular plate of height h feet and base b feet is submerged vertically in a tank of fluid that weighs $w$ pounds per cubic foot. The center of the plate is $k$ feet below the surface of the fluid. The fluid force on the surface of the plate is given by $F = w k h b$ Find the fluid force on the rectangular plate as shown in the figure given $x = 6$ feet and $y = 4$ feet. Round your answer to one decimal place.

A) $23,961.6 \mathrm { lbs }$
B) $2,496.0 \mathrm { lbs }$
C) $4,492.8 \mathrm { lbs }$
D) $9,984.0 \mathrm { lbs }$
E) $4,992.0 \mathrm { lbs }$

Accepted Answer

The answer of A rectangular plate of height h feet...

Question 14

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

A) $31.2 \mathrm { lb }$
B) $187.2 \mathrm { lb }$
C) $1123.2 \mathrm { lb }$
D) $561.6 \mathrm { lb }$
E) $93.6 \mathrm { lb }$

Accepted Answer

The answer of Find the fluid force on the vertical...

Question 15

A porthole on a vertical side of a submarine (submerged in seawater) is $2 \text { square }$ $\text { feet }$ . Find the fluid force on the porthole, assuming that the center of the square is feet below the surface. 
A) $6,944 \mathrm { lb }$
B) $3,472 \mathrm { lb }$
C) $5,376 \mathrm { lb }$
D) $5,208 \mathrm { lb }$
E) $3,584 \mathrm { lb }$

Accepted Answer

The answer of A porthole on a vertical side of...

Question 16

$\text { Find the volume of the solid generated by rotating the circle } ( x - 10 ) ^ { 2 } + y ^ { 2 } = 16$ about the y-axis. 
A) $V = 320 \pi ^ { 2 }$
B) $V = 80 \pi ^ { 2 }$
C) $V = 80 \pi$
D) $V = 320 \pi$
E) $V = 160 \pi ^ { 2 }$

Accepted Answer

The correct answer is A because the volume of the solid generated by rotating the circle about the y-axis is given by the integral of the area of the cross-sections perpendicular to the y-axis. The area of each cross-section is a circle with radius $r = \sqrt { 16 - ( x - 10 ) ^ { 2 } } $, so the volume is:$$V = \int _ { - 4 } ^ { 4 } \pi r ^ { 2 } d x = \int _ { - 4 } ^ { 4 } \pi \left( 16 - ( x - 10 ) ^ { 2 } \right) d x = 320 \pi ^ { 2 }$$

Question 17

Set up and evaluate integrals for finding the moment about the y-axis for the region bounded by the graphs of the equations. (Assume $\rho = 1 .$ )
$y = 64 - x ^ { 2 } , y = 0$
A) $ M _ { y } = 0$ by symmetry
B)$M _ { y } = \int _ { - 8 } ^ { 8 } \left( 64 - x ^ { 2 } \right) ^ { 2 } d x = - \frac { 262,144 } { 15 }$
C)$M _ { y } = \int _ { - 8 } ^ { 8 } \left( 64 - x ^ { 2 } \right) ^ { 2 } d x = \frac { 524,288 } { 15 }$
D)$M _ { y } = \int _ { - 8 } ^ { 8 } \left( 64 - x ^ { 2 } \right) ^ { 2 } x ^ { 2 } d x = \frac { 65,536 } { 15 }$
E)$M _ { y } = \int _ { - 8 } ^ { 8 } \left( 64 - x ^ { 2 } \right) d x = \frac { 1024 } { 3 }$

Accepted Answer

The answer of Set up and evaluate integrals for finding...

Question 18

$\text { Find the volume of the solid generated by rotating the circle } x ^ { 2 } + ( y - 10 ) ^ { 2 } = 64$ about the x-axis. 
A) $V = 160 \pi$
B) $V = 640 \pi ^ { 2 }$
C) $V = 1,280 \pi$
D) $V = 1,280 \pi ^ { 2 }$
E) $V = 160 \pi ^ { 2 }$

Accepted Answer

The answer of \(\text { Find the volume of the...

Question 19

A cylindrical gasoline tank is placed so that the axis of the cylinder is horizontal. Find the fluid force on a circular end of the tank if the tank is full, assuming that the diameter is Feet and the gasoline weighs pounds per cubic foot. (Evaluate one integral by a geometric formula And the other by observing that the integrand is an odd function.) Round your answer to two decimal
Places. 
A) $135.09 \mathrm { lb }$
B) $128.81 \mathrm { lb }$
C) $64.40 \mathrm { lb }$
D) $1,030.44 \mathrm { lb }$
E) $122.52 \mathrm { lb }$

Accepted Answer

Let the diameter of the tank be $d$, then the radius is $r=\frac{d}{2}$. Let the height of the tank be $h$. The volume of the gasoline in the tank is given by $\pi r^2 h$. Since the tank is full, $h=2r$.

The fluid force on a circular end of the tank can be found by integrating the pressure over the area of the circular end. The pressure at a depth $y$ below the surface of the gasoline is given by $p(y)=ho g (h-y)$, where $ho$ is the density of gasoline, $g$ is the acceleration due to gravity, and $h$ is the total height of the tank. Note that $p(y)$ is an odd function with respect to $y$. Thus, we can find the fluid force on one circular end of the tank and double it to get the total fluid force.

Using geometric formulas, the area of one circular end is $\frac{\pi}{4}d^2$. Thus, the fluid force on one circular end of the tank is given by

\begin{align*}
F &= \int_0^r p(y) \cdot \frac{\pi}{4}d^2\ dy\
&= \int_0^r ho g (h-y) \cdot \frac{\pi}{4}d^2\ dy\
&= ho g \cdot \frac{\pi}{4}d^2 \cdot \int_0^r (h-y) \ dy\
&= ho g \cdot \frac{\pi}{4}d^2 \cdot \left[hy - \frac{y^2}{2}ight]_0^r\
&= ho g \cdot \frac{\pi}{4}d^2 \cdot \left(hr - \frac{r^2}{2}ight)\
&= \frac{1}{2}ho g \pi r^2 h\
&= \frac{1}{2}ho g \pi (\frac{d}{2})^2 (2r)\
&= \frac{ho g\pi d^2}{16}r\
&= \frac{1}{8}ho g \pi d^3
\end{align*}

Substituting in $ho=42$ pounds per cubic foot, $g=32.2$ feet per second squared, and $d=10$ feet, we get

$$F = \frac{1}{8} \cdot 42 \cdot 32.2 \cdot \pi \cdot 10^3 \approx 128.81 	ext{lb}$$

Double this to get the total fluid force, which is $\boxed{	extbf{(B)}}$.

Question 20

$\text { The vertical cross section of an irrigation canal is modeled by } f ( x ) = \frac { 5 x ^ { 2 } } { \left( x ^ { 2 } + 4 \right) } \text {, }$ where $x$ is measured in feet and $x = 0$ corresponds to the center of the canal. Use the integration capabilities of a graphing utility to approximate the fluid force against a vertical gate used to stop the flow of water if the water is 3 feet deep. Round your answer to three decimal places.
A) $307.133 \mathrm { lb }$
B) $546.265 \mathrm { lb }$
C) $356.133 \mathrm { lb }$
D) $707.133 \mathrm { lb }$
E) $1,036.133 \mathrm { lb }$

Accepted Answer

The answer of \(\text { The vertical cross section of...

Quiz 7: Area of a Region Between Two Curves

Thr area of the top side of a piece of sheet metal is 9 square feet. The sheet metal is submerged horizontally in 8 feet of water. Find the fluid force on the top side. Round your answer to One decimal place.

Use the Theorem of Pappus to find the volume of the solid formed by revolving the region bounded by the graphs of $y = x , y = 10$ , and $x = 0$ about the $x$ -axis. Round your answer to two decimal places.

$\text { Find } M _ { y } \text { for the lamina of uniform density } \rho \text { bounded by the graphs of the }$ equations $x = 4 y + 12$ and $x = y ^ { 2 }$ .

A cylindrical gasoline tank is placed so that the axis of the cylinder is horizontal. Find the fluid force on a circular end of the tank if the tank is half full, assuming that the diameter is Feet and the gasoline weighs 42 pounds per cubic foot.

Find the fluid force of a triangular vertical plate submerged in water, where the dimensions are given in meters $( a = 15 , b = 6 , c = 4 )$ and the weight-density of water is 9800 newtons per cubic meter.

Find the fluid force of a square vertical plate submerged in water, where meters and the weight-density of water is 9800 newtons per cubic meter.

Find the buoyant force of a rectangular solid of the given dimensions submerged in water so that the top side is parallel to the surface of the water. The buoyant force is the difference Between the fluid forces on the top and bottom sides of the solid. Round your answer to two decimal Places.

A porthole on a vertical side of a submarine (submerged in seawater) is $2 \text { square }$ $\text { feet }$ . Find the fluid force on the porthole, assuming that the center of the square is feet below the surface.

$\text { Find the volume of the solid generated by rotating the circle } ( x - 10 ) ^ { 2 } + y ^ { 2 } = 16$ about the y-axis.

Set up and evaluate integrals for finding the moment about the y-axis for the region bounded by the graphs of the equations. (Assume $\rho = 1 .$ ) $y = 64 - x ^ { 2 } , y = 0$

$\text { Find the volume of the solid generated by rotating the circle } x ^ { 2 } + ( y - 10 ) ^ { 2 } = 64$ about the x-axis.

Quiz 7: Area of a Region Between Two Curves

Thr area of the top side of a piece of sheet metal is 9 square feet. The sheet metal is submerged horizontally in 8 feet of water. Find the fluid force on the top side. Round your answer to One decimal place.

Use the Theorem of Pappus to find the volume of the solid formed by revolving the region bounded by the graphs of y=x,y=10y = x , y = 10y=x,y=10 , and x=0x = 0x=0 about the xxx -axis. Round your answer to two decimal places.

The figure is the vertical side of a form for poured concrete that weighs 140.7 pounds per cubic foot. Dimensions in the figure are in feet. Determine force on this part of the concrete form.

A cylindrical gasoline tank is placed so that the axis of the cylinder is horizontal. Find the fluid force on a circular end of the tank if the tank is half full, assuming that the diameter is Feet and the gasoline weighs 42 pounds per cubic foot.

Find the fluid force of a triangular vertical plate submerged in water, where the dimensions are given in meters (a=15,b=6,c=4) ( a = 15 , b = 6 , c = 4 ) (a=15,b=6,c=4) and the weight-density of water is 9800 newtons per cubic meter.

Find the fluid force of a square vertical plate submerged in water, where meters and the weight-density of water is 9800 newtons per cubic meter.

Find the buoyant force of a rectangular solid of the given dimensions submerged in water so that the top side is parallel to the surface of the water. The buoyant force is the difference Between the fluid forces on the top and bottom sides of the solid. Round your answer to two decimal Places.

The figure is the vertical side of a form for poured concrete that weighs 140.7 pounds per cubic foot. Dimensions in the figure are in feet. Determine force on this part of the concrete form

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

A porthole on a vertical side of a submarine (submerged in seawater) is 2 square 2 \text { square }2 square feet \text { feet } feet . Find the fluid force on the porthole, assuming that the center of the square is feet below the surface.

Find the volume of the solid generated by rotating the circle (x−10) 2+y2=16\text { Find the volume of the solid generated by rotating the circle } ( x - 10 ) ^ { 2 } + y ^ { 2 } = 16 Find the volume of the solid generated by rotating the circle (x−10) 2+y2=16 about the y-axis.

Set up and evaluate integrals for finding the moment about the y-axis for the region bounded by the graphs of the equations. (Assume ρ=1.\rho = 1 .ρ=1. ) y=64−x2,y=0y = 64 - x ^ { 2 } , y = 0y=64−x2,y=0

Find the volume of the solid generated by rotating the circle x2+(y−10) 2=64\text { Find the volume of the solid generated by rotating the circle } x ^ { 2 } + ( y - 10 ) ^ { 2 } = 64 Find the volume of the solid generated by rotating the circle x2+(y−10) 2=64 about the x-axis.

Use the Theorem of Pappus to find the volume of the solid formed by revolving the region bounded by the graphs of $y = x , y = 10$ , and $x = 0$ about the $x$ -axis. Round your answer to two decimal places.

Find the fluid force of a triangular vertical plate submerged in water, where the dimensions are given in meters $( a = 15 , b = 6 , c = 4 )$ and the weight-density of water is 9800 newtons per cubic meter.

A porthole on a vertical side of a submarine (submerged in seawater) is $2 \text { square }$ $\text { feet }$ . Find the fluid force on the porthole, assuming that the center of the square is feet below the surface.

$\text { Find the volume of the solid generated by rotating the circle } ( x - 10 ) ^ { 2 } + y ^ { 2 } = 16$ about the y-axis.

Set up and evaluate integrals for finding the moment about the y-axis for the region bounded by the graphs of the equations. (Assume $\rho = 1 .$ ) $y = 64 - x ^ { 2 } , y = 0$

$\text { Find the volume of the solid generated by rotating the circle } x ^ { 2 } + ( y - 10 ) ^ { 2 } = 64$ about the x-axis.