Question 1

A tank contains water. The end of the tank is vertical and has the shape below. Find the hydrostatic force against the end of the tank.

Accepted Answer

The answer of A tank contains water. The end of...

Question 2

A 100-foot length of steel chain weighing 5 lb/ft is dangling from the drum of a winch.(a) How much work is required to wind the chain onto the drum?
(b) How much work is required to wind the first 30 feet of chain onto the drum?

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Question 3

A tank contains water. The end of the tank is vertical and has the shape below. Find the hydrostatic force against the end of the tank.

Accepted Answer

The answer of A tank contains water. The end of...

Question 4

A tank contains water. The end of the tank is vertical and has the shape below. Find the hydrostatic force against the end of the tank.

Accepted Answer

The answer of A tank contains water. The end of...

Question 5

The tank in the figure below is full of water with a density 62.5 lb/ft³. How much work is required to empty the tank by pumping water to a point 4 feet above the top of the tank?

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Question 6

A tank 5 feet long has cross-sections n the shape of a parabola $y = x ^ { 2 } , \text { for } - 2 \leq x \leq 2$ (where x and y are in feet). Suppose that the tank is filled to a depth of 3 feet with liquid weighing 15 lb/ft³. How much work is required to empty the tank by pumping the liquid over the edge of the tank?

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Question 7

How much work is done to bring a load of ore to the surface if a miner uses a cable weighing 2 lb/ft to haul a 100 lb bucket of ore up a mine shaft 800 feet deep?

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Question 8

Determine the centroid of the region bounded by the equation $y ^ { 2 } - 9 x = 0$ in the first quadrant between x = 1 and x = 4.

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Question 9

Find the total hydraulic force on a dam in the shape of an equilateral triangle with one vertex pointing down, if the side of the triangle is 100 feet and the water is even with the top.

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Question 10

Find the centroid of the region bounded by $y = \cos x , y = 0 \text { and } 0 \leq x \leq \frac { \pi } { 2 }$ .

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Question 11

City B (population 4,000) is 10 miles north of City A (population 6,000), and City C (population 5,000) is 30 miles east of City B. Where is the best place to locate a super market serving the people in these cities?

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Question 12

A swimming pool 5 m wide, 10 m long, and 3 m deep is filled with seawater of density 1030 kg/m³ to a depth of 2.5 m. Find (a) the hydrostatic pressure at the bottom of the pool.(b) the hydrostatic force on the bottom.(c) the hydrostatic force on one end of the pool.

Accepted Answer

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Question 13

A hemispherical tank with radius 8 feet is filled with water to a depth of 6 feet. Find the work required to empty the tank by pumping the water to the top of the tank.

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Question 14

A flat plate of negligible thickness is in the shape of a right triangle with base 5 ft and height 10 ft. The plate is submerged in a tank of water. Find the force on the face of the plate under the following conditions: (Use 62.4 lb/ft³ as the density of water.) (a) The plate is submerged horizontally so that it rests flat on the bottom of the tank at a depth of 14 ft.(b) The plate is submerged vertically, base edge up and base at a depth of 3 ft.

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Question 15

Calculate the center of mass of a lamina with the given density and shape:

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Question 16

Find the center of mass of the lamina of uniform density $\delta$ bounded by $y = \sqrt { 1 - x ^ { 2 } }$ and x-axis.

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Question 17

Find the centroid of the region bounded by $y = 1 - x ^ { 2 } \text { and } y = 0$

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Question 18

Find the centroid of the region bounded by $y = \sin x , y = 0 \text { and } 0 \leq x \leq \pi$ .

Accepted Answer

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Question 19

Suppose a hemispherical tank with radius 10 feet is filled with liquid whose density is 62 pounds per cubic foot. Find the work required to pump all the liquid out through the top of the tank.

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Question 20

Two people weighing 100 lb and 180 lb are at opposite ends of a seesaw 14 ft long. Where should the point of support be placed so that the seesaw will balance?

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Quiz 6: Applications of Integration

A tank contains water. The end of the tank is vertical and has the shape below. Find the hydrostatic force against the end of the tank.

A 100-foot length of steel chain weighing 5 lb/ft is dangling from the drum of a winch.(a) How much work is required to wind the chain onto the drum? (b) How much work is required to wind the first 30 feet of chain onto the drum?

A tank contains water. The end of the tank is vertical and has the shape below. Find the hydrostatic force against the end of the tank.

A tank contains water. The end of the tank is vertical and has the shape below. Find the hydrostatic force against the end of the tank.

The tank in the figure below is full of water with a density 62.5 lb/ft3. How much work is required to empty the tank by pumping water to a point 4 feet above the top of the tank?

How much work is done to bring a load of ore to the surface if a miner uses a cable weighing 2 lb/ft to haul a 100 lb bucket of ore up a mine shaft 800 feet deep?

Determine the centroid of the region bounded by the equation y2−9x=0y ^ { 2 } - 9 x = 0y2−9x=0 in the first quadrant between x = 1 and x = 4.

Find the total hydraulic force on a dam in the shape of an equilateral triangle with one vertex pointing down, if the side of the triangle is 100 feet and the water is even with the top.

Find the centroid of the region bounded by y=cos⁡x,y=0 and 0≤x≤π2y = \cos x , y = 0 \text { and } 0 \leq x \leq \frac { \pi } { 2 }y=cosx,y=0 and 0≤x≤2π​ .

City B (population 4,000) is 10 miles north of City A (population 6,000), and City C (population 5,000) is 30 miles east of City B. Where is the best place to locate a super market serving the people in these cities?

A swimming pool 5 m wide, 10 m long, and 3 m deep is filled with seawater of density 1030 kg/m3 to a depth of 2.5 m. Find (a) the hydrostatic pressure at the bottom of the pool.(b) the hydrostatic force on the bottom.(c) the hydrostatic force on one end of the pool.

A hemispherical tank with radius 8 feet is filled with water to a depth of 6 feet. Find the work required to empty the tank by pumping the water to the top of the tank.

Calculate the center of mass of a lamina with the given density and shape:

Find the center of mass of the lamina of uniform density δ\deltaδ bounded by y=1−x2y = \sqrt { 1 - x ^ { 2 } }y=1−x2​ and x-axis.

Find the centroid of the region bounded by y=1−x2 and y=0y = 1 - x ^ { 2 } \text { and } y = 0y=1−x2 and y=0

Find the centroid of the region bounded by y=sin⁡x,y=0 and 0≤x≤πy = \sin x , y = 0 \text { and } 0 \leq x \leq \piy=sinx,y=0 and 0≤x≤π .

Suppose a hemispherical tank with radius 10 feet is filled with liquid whose density is 62 pounds per cubic foot. Find the work required to pump all the liquid out through the top of the tank.

Two people weighing 100 lb and 180 lb are at opposite ends of a seesaw 14 ft long. Where should the point of support be placed so that the seesaw will balance?

The tank in the figure below is full of water with a density 62.5 lb/ft³. How much work is required to empty the tank by pumping water to a point 4 feet above the top of the tank?

Determine the centroid of the region bounded by the equation $y ^ { 2 } - 9 x = 0$ in the first quadrant between x = 1 and x = 4.

Find the centroid of the region bounded by $y = \cos x , y = 0 \text { and } 0 \leq x \leq \frac { \pi } { 2 }$ .

A swimming pool 5 m wide, 10 m long, and 3 m deep is filled with seawater of density 1030 kg/m³ to a depth of 2.5 m. Find (a) the hydrostatic pressure at the bottom of the pool.(b) the hydrostatic force on the bottom.(c) the hydrostatic force on one end of the pool.

Find the center of mass of the lamina of uniform density $\delta$ bounded by $y = \sqrt { 1 - x ^ { 2 } }$ and x-axis.

Find the centroid of the region bounded by $y = 1 - x ^ { 2 } \text { and } y = 0$

Find the centroid of the region bounded by $y = \sin x , y = 0 \text { and } 0 \leq x \leq \pi$ .