Question 1

An aquarium 1 foot high, 1 foot wide, and 2 feet long is filled with water. For simplicity, take the density of water to be 60 lb/ft³. Find the hydrostatic force in pound on one of the 1 foot by 2 foot sides of the aquarium. A)336 B)30 C)168 D)240 E)60 F)120 G)28 H)12

Accepted Answer

The answer of An aquarium 1 foot high, 1 foot...

Question 2

Find the y-coordinate of the centroid of the region bounded by the curves $y = x ^ { 2 } , y = 1$ 
A)0.85
B)0.70
C)0.60
D)0.50
E)0.75
F)0.55
G)0.80
H)0.65

Accepted Answer

The answer of Find the y-coordinate of the centroid of...

Question 3

Find the center of mass of the linear system $m _ { 1 } = 2 , x _ { 1 } = - 3 , m _ { 2 } = 8$ $x _ { 2 } = - 1 , m _ { 3 } = 3$ $x _ { 3 } = 1 , m _ { 4 } = 5 , x _ { 4 } = 4$ 
A)9
B) $\frac { 1 } { 2 }$ 
C) $- \frac { 1 } { 2 }$ 
D)0
E) $\frac { 2 } { 3 }$ 
F) $- \frac { 2 } { 3 }$ 
G) $- 9$ 
H)18

Accepted Answer

The answer of Find the center of mass of the...

Question 4

Find the center of mass of the system $m _ { 1 } = 5 , P _ { 1 } ( - 3,1 ) , m _ { 2 } = 9$ $P _ { 2 } = ( - 1 , - 1 ) , m _ { 3 } = 6$ $P _ { 3 } = ( 1,1 ) , m _ { 4 } = 8 , P _ { 4 } ( 3 , - 2 )$ 
A) $\left( \frac { 3 } { 14 } , - \frac { 1 } { 2 } \right)$ 
B) $\left( - \frac { 1 } { 2 } , \frac { 3 } { 14 } \right)$ 
C) $\left( \frac { 3 } { 14 } , \frac { 1 } { 2 } \right)$ 
D) $\left( - \frac { 3 } { 14 } , \frac { 1 } { 2 } \right)$ 
E) $( 6 , - 14 )$ 
F) $( - 14,6 )$ 
G) $( 14 , - 6 )$ 
H) $( - 14 , - 6 )$

Accepted Answer

The center of mass (COM) of a system of particles is given by the formula $\left( \frac{\sum m_i x_i}{\sum m_i}, \frac{\sum m_i y_i}{\sum m_i} \right)$. For the given system, the x-coordinate of the COM is $\frac{5(-3) + 9(-1) + 6(1) + 8(3)}{5 + 9 + 6 + 8} = \frac{-15 - 9 + 6 + 24}{28} = \frac{6}{28} = \frac{3}{14}$, and the y-coordinate is $\frac{5(1) + 9(-1) + 6(1) + 8(-2)}{28} = \frac{5 - 9 + 6 - 16}{28} = \frac{-14}{28} = -\frac{1}{2}$. Therefore, the center of mass is $\left( \frac{3}{14}, -\frac{1}{2} \right)$.

Question 5

A force of 8 dynes is required to stretch a spring from its natural length of 10 cm to a length of 15 cm. How much work is done
(a) in stretching the spring to a length of 25 cm?
(b) in stretching the spring from a length of 20 cm to a length of 25 cm?

Accepted Answer

The answer of A force of 8 dynes is required...

Question 6

A right circular cylinder tank of height 1 foot and radius 1 foot is full of water. Taking the density of water to be a nice round 60 pounds per cubic foot, how much work in foot-pounds is required to pump all of the water up and over the top of the tank?
A)8 $\pi$ 
B)24 $\pi$ 
C)16 $\pi$ 
D)6 $\pi$ 
E)5 $\pi$ 
F)4 $\pi$ 
G)30 $\pi$ 
H)18 $\pi$

Accepted Answer

The answer of A right circular cylinder tank of height...

Question 7

A swimming pool 24 feet long and 15 feet wide has a bottom that is an inclined plane, the shallow end having a depth of 3 feet, and the deep end 10 feet. The pool is filled with water. For simplicity, take the density of water to be 60 lb/ft³. Find the hydrostatic force in pounds on the bottom of the pool. A)145,750 B)146,250 C)147,000 D)147,250 E)146,500 F)147,500 G)146,750 H)146,000

Accepted Answer

The hydrostatic force on the bottom of the pool can be calculated by multiplying the area of the bottom by the average depth and then by the density of water. The area of the bottom is $24 \times 15 = 360 \, \text{ft}^2$. The average depth is $\frac{3 + 10}{2} = 6.5 \, \text{ft}$. Therefore, the hydrostatic force is $360 \times 6.5 \times 60 = 140,400 \, \text{lb}$. However, since this result does not match any of the options and considering the nature of the question, it seems there was a miscalculation in the explanation provided. The correct approach to calculate the hydrostatic force on the bottom of the pool involves considering the volume of water and the density, but the calculation provided was incorrect. The correct calculation involves the volume of water in the pool, which is not directly given by the average depth method mentioned. Given the error in the explanation, let's correct the approach:The volume of water in the pool can be considered as the volume of a trapezoidal prism, but since the question asks for the force on the bottom, we focus on the pressure due to the water directly above the bottom surface, which is uniform across the entire bottom. The force is the weight of the water, which is the product of the volume and the density of water.Given the mistake in the initial explanation, the correct step involves calculating the weight of the water:$$ \text{Volume of water} = \text{Area of bottom} \times \text{Average depth} $$However, the average depth approach was incorrectly applied to the force calculation instead of focusing on the weight of the water. The correct calculation should directly use the density of water and the gravitational force to find the hydrostatic force without needing the average depth, as the force on the bottom is due to the total weight of the water:$$ \text{Hydrostatic force} = \text{Volume of water} \times \text{Density of water} $$Given the error in the provided explanation and the lack of correct calculation steps for the options provided, the correct answer should be recalculated based on the correct formula for hydrostatic force:$$ \text{Hydrostatic force} = \text{Volume} \times \text{Density} $$Since the volume and specific calculation steps were not correctly provided in the explanation, and the initial response was incorrect, the correct answer should be recalculated with accurate information. Without the correct volume calculation, the specific option cannot be accurately determined from the provided explanation.

Question 8

Find the center of mass of the linear system $m _ { 1 } = 1 , m _ { 2 } = 9 , m _ { 3 } = 6$ $x _ { 1 } = - 3 , \quad x _ { 2 } = - 1$ $x _ { 3 } = 1$ .
A) $- \frac { 3 } { 8 }$ 
B) $\frac { 3 } { 8 }$ 
C)6
D) $\frac { 2 } { 3 }$ 
E) $- \frac { 3 } { 4 }$ 
F) $\frac { 3 } { 4 }$ 
G) $- 6$ 
H) $- \frac { 2 } { 3 }$

Accepted Answer

The answer of Find the center of mass of the...

Question 9

A right circular cylinder tank of height 1 foot and radius 1 foot is full of water. Taking the density of water to be a nice round 60 pounds per cubic foot, find the hydrostatic force in pounds on the side of the tank.
A)30 $\pi$ 
B)60
C)240 $\pi$ 
D)240
E)30
F)120 $\pi$ 
G)60 $\pi$ 
H)120

Accepted Answer

The hydrostatic force on the side of the tank is calculated by integrating the pressure over the surface area. The pressure at a depth $ y $ is $ 60y $ (since pressure = density x depth). The force is the integral of pressure times the differential area, which results in $ 60 \pi \int_0^1 y \, dy = 60 \pi \left[ \frac{y^2}{2} \right]_0^1 = 30 \pi $.

Question 10

A rope 100 feet long weighing 2 pounds per foot hangs over the edge of a building 100 feet tall. How much work in foot-pounds is done in pulling the rope to the top of the building?
A)7500
B)5000
C)1000
D)500
E)10,000
F)750
G)1250
H)12,500

Accepted Answer

The work done in pulling the rope to the top is calculated by integrating the weight of the rope over the distance it is lifted. The weight of the rope is 2 pounds per foot, and the center of mass of the rope is at 50 feet (halfway up). Thus, the work done is 2 pounds/foot * 100 feet * 50 feet = 10,000 foot-pounds.

Question 11

The dye dilution method is used to measure cardiac output with 6 mg of dye. The dye concentrations, in mg/L, are modeled by $c ( t ) = \frac { 1 } { 15 } t ( 15 - t ) , \quad 0 \leq t \leq 15$ , where t is measured in seconds. Find the cardiac output.

Accepted Answer

The answer of The dye dilution method is used to...

Question 12

The following table shows the relationship between price and demand for milk produced in a large dairy.$$\begin{array} { | l | c | c | c | c | c | c | c | } 
\hline q \text { (billions of pounds of milk per year) } & 45 & 50 & 55 & 60 & 65 & 70 & 75 \
\hline p \text { (price in dollars per pound) } & 1.00 & 0.90 & 0.80 & 0.70 & 0.60 & 0.50 & 0.40 \
\hline
\end{array}$$
Determine the consumer's surplus when the sales quality is 65 billion pounds of milk in a year. Illustrate your answer by drawing the corresponding demand curve and the identifying the consumer's surplus as a region.

Accepted Answer

The answer of The following table shows the relationship between...

Question 13

Find the work (in ft-lb) done in raising 500 lb of ore from a mine that is 1000 ft deep. Assume that the cable used to raise the ore weighs 2 lb/ft. A) $5 \times 10 ^ { 5 }$ B) $2.5 \times 10 ^ { 6 }$ C) $2 \times 10 ^ { 6 }$ D) $6 \times 10 ^ { 5 }$ E)10⁶ F) $1.5 \times 10 ^ { 6 }$ G) $7 \times 10 ^ { 5 }$ H) $4 \times 10 ^ { 5 }$

Accepted Answer

The work done is the sum of the work to lift the ore and the work to lift the cable. The work to lift the ore is $500 \, \text{lb} \times 1000 \, \text{ft} = 5 \times 10^5 \, \text{ft-lb}$. The work to lift the cable is $\int_0^{1000} 2x \, dx = 10^6 \, \text{ft-lb}$. Total work is $5 \times 10^5 + 10^6 = 1.5 \times 10^6 \, \text{ft-lb}$.

Question 14

A force of 20 pounds stretches a spring $\frac { 3 } { 4 }$ foot beyond its natural length. Find the work done in stretching the spring 1 foot.
A)20 ft-lb
B) $\frac { 20 } { 3 }$ ft-lb
C) $\frac { 80 } { 3 }$ ft-lb
D) $\frac { 40 } { 3 }$ ft-lb
E) $\frac { 40 } { 9 }$ ft-lb
F) $\frac { 80 } { 9 }$ ft-lb
G) $\frac { 40 } { 6 }$ ft-lb
H) $\frac { 80 } { 6 }$ ft-lb

Accepted Answer

The answer of A force of 20 pounds stretches a...

Question 15

Find the x-coordinate of the centroid of the region bounded by the graphs $y = \sqrt [ 3 ] { x } , \quad x = 8$ and the x-axis.
A)8
B) $\frac { 16 } { 7 }$ 
C) $\frac { 7 } { 16 }$ 
D) $\frac { 224 } { 9 }$ 
E)4

F) $\frac { 32 } { 7 }$ 
G) $\frac { 7 } { 32 }$ 
H) $\frac { 2 } { 3 }$

Accepted Answer

The x-coordinate of the centroid is given by $\frac{1}{A} \int_{a}^{b} x f(x) \, dx$, where $A$ is the area under the curve. For $y = \sqrt[3]{x}$ from $x = 0$ to $x = 8$, the area $A = \int_{0}^{8} \sqrt[3]{x} \, dx = \frac{24}{5}$. The x-coordinate of the centroid is $\frac{1}{A} \int_{0}^{8} x \sqrt[3]{x} \, dx = \frac{32}{7}$.

Question 16

If $\frac { 15 } { 4 }$ ft-lb of work is needed to stretch a spring a length $\frac { 1 } { 2 }$ ft beyond its natural length, find the spring constant.

Accepted Answer

The answer of If \(\frac { 15 } { 4...

Question 17

A spring stretches 1 foot beyond its natural position under a force of 100 points. How much work in foot-pounds is done in stretching it 3 feet beyond its natural position?
A)600
B)30
C)1500
D)450
E)100
F)900
G)150
H)300

Accepted Answer

The work done on a spring is calculated using Hooke's Law, W = (1/2)kx^2. Given that a force of 100 pounds stretches the spring 1 foot, the spring constant k is 100. To stretch it 3 feet, the work done is (1/2)(100)(3^2) = 450 foot-pounds.

Question 18

A gate in an irrigation canal is in the form of a trapezoid 3 feet wide at the bottom, 5 feet wide at the top, with height equal to 2 feet. It is placed vertically in the canal with the water extending to its top. For simplicity, take the density of water to be 60 lb/ft³. Find the hydrostatic force in pounds on the gate. A)360 B)380 C)440 D)420 E)400 F)460 G)500 H)480

Accepted Answer

The answer of A gate in an irrigation canal is...

Question 19

Find the x-coordinate x at the centroid of the region bounded by the x-axis and the lines $y = 3 x , \quad x = 2$ 
A) $\frac { 4 } { 3 }$ 
B) $\frac { 7 } { 6 }$ 
C) $\frac { 10 } { 7 }$ 
D) $\frac { 5 } { 3 }$ 
E) $\frac { 11 } { 7 }$ 
F) $\frac { 11 } { 6 }$ 
G) $\frac { 5 } { 4 }$ 
H) $\frac { 7 } { 4 }$

Accepted Answer

The x-coordinate of the centroid of a region bounded by the x-axis, y = 3x, and x = 2 is found using the formula for the centroid $ \bar{x} = \frac{1}{A} \int_{a}^{b} x \cdot f(x) dx $, where A is the area of the region, and f(x) is the function defining the curve. Here, $ f(x) = 3x $, and the limits of integration are from 0 to 2. The area $ A = \int_{0}^{2} 3x dx = \frac{3}{2}x^2 \Big|_0^2 = 6 $. Thus, $ \bar{x} = \frac{1}{6} \int_{0}^{2} x \cdot 3x dx = \frac{1}{6} \int_{0}^{2} 3x^2 dx = \frac{1}{6} \cdot \frac{3}{3}x^3 \Big|_0^2 = \frac{1}{6} \cdot 8 = \frac{4}{3} $.

Question 20

A force of 10 pounds is required to stretch a spring from its natural length of 8 inches to a length of 10 inches. How much work is done in stretching the spring to a length of 12 inches?

Accepted Answer

The answer of A force of 10 pounds is required...

Quiz 6: Applications of Integration

An aquarium 1 foot high, 1 foot wide, and 2 feet long is filled with water. For simplicity, take the density of water to be 60 lb/ft3 . Find the hydrostatic force in pound on one of the 1 foot by 2 foot sides of the aquarium.

Find the y-coordinate of the centroid of the region bounded by the curves y=x2,y=1y = x ^ { 2 } , y = 1y=x2,y=1

Find the center of mass of the linear system m1=2,x1=−3,m2=8m _ { 1 } = 2 , x _ { 1 } = - 3 , m _ { 2 } = 8m1​=2,x1​=−3,m2​=8 x2=−1,m3=3x _ { 2 } = - 1 , m _ { 3 } = 3x2​=−1,m3​=3 x3=1,m4=5,x4=4x _ { 3 } = 1 , m _ { 4 } = 5 , x _ { 4 } = 4x3​=1,m4​=5,x4​=4

A force of 8 dynes is required to stretch a spring from its natural length of 10 cm to a length of 15 cm. How much work is done (a) in stretching the spring to a length of 25 cm? (b) in stretching the spring from a length of 20 cm to a length of 25 cm?

A right circular cylinder tank of height 1 foot and radius 1 foot is full of water. Taking the density of water to be a nice round 60 pounds per cubic foot, how much work in foot-pounds is required to pump all of the water up and over the top of the tank?

Find the center of mass of the linear system m1=1,m2=9,m3=6m _ { 1 } = 1 , m _ { 2 } = 9 , m _ { 3 } = 6m1​=1,m2​=9,m3​=6 x1=−3,x2=−1x _ { 1 } = - 3 , \quad x _ { 2 } = - 1x1​=−3,x2​=−1 x3=1x _ { 3 } = 1x3​=1 .

A right circular cylinder tank of height 1 foot and radius 1 foot is full of water. Taking the density of water to be a nice round 60 pounds per cubic foot, find the hydrostatic force in pounds on the side of the tank.

A rope 100 feet long weighing 2 pounds per foot hangs over the edge of a building 100 feet tall. How much work in foot-pounds is done in pulling the rope to the top of the building?

Find the work (in ft-lb) done in raising 500 lb of ore from a mine that is 1000 ft deep. Assume that the cable used to raise the ore weighs 2 lb/ft.

A force of 20 pounds stretches a spring 34\frac { 3 } { 4 }43​ foot beyond its natural length. Find the work done in stretching the spring 1 foot.

Find the x-coordinate of the centroid of the region bounded by the graphs y=x3,x=8y = \sqrt [ 3 ] { x } , \quad x = 8y=3x​,x=8 and the x-axis.

If 154\frac { 15 } { 4 }415​ ft-lb of work is needed to stretch a spring a length 12\frac { 1 } { 2 }21​ ft beyond its natural length, find the spring constant.

A spring stretches 1 foot beyond its natural position under a force of 100 points. How much work in foot-pounds is done in stretching it 3 feet beyond its natural position?

Find the x-coordinate x at the centroid of the region bounded by the x-axis and the lines y=3x,x=2y = 3 x , \quad x = 2y=3x,x=2

A force of 10 pounds is required to stretch a spring from its natural length of 8 inches to a length of 10 inches. How much work is done in stretching the spring to a length of 12 inches?

An aquarium 1 foot high, 1 foot wide, and 2 feet long is filled with water. For simplicity, take the density of water to be 60 lb/ft³. Find the hydrostatic force in pound on one of the 1 foot by 2 foot sides of the aquarium.

Find the y-coordinate of the centroid of the region bounded by the curves $y = x ^ { 2 } , y = 1$

Find the center of mass of the linear system $m _ { 1 } = 2 , x _ { 1 } = - 3 , m _ { 2 } = 8$ $x _ { 2 } = - 1 , m _ { 3 } = 3$ $x _ { 3 } = 1 , m _ { 4 } = 5 , x _ { 4 } = 4$

Find the center of mass of the linear system $m _ { 1 } = 1 , m _ { 2 } = 9 , m _ { 3 } = 6$ $x _ { 1 } = - 3 , \quad x _ { 2 } = - 1$ $x _ { 3 } = 1$ .

A force of 20 pounds stretches a spring $\frac { 3 } { 4 }$ foot beyond its natural length. Find the work done in stretching the spring 1 foot.

Find the x-coordinate of the centroid of the region bounded by the graphs $y = \sqrt [ 3 ] { x } , \quad x = 8$ and the x-axis.

If $\frac { 15 } { 4 }$ ft-lb of work is needed to stretch a spring a length $\frac { 1 } { 2 }$ ft beyond its natural length, find the spring constant.

Find the x-coordinate x at the centroid of the region bounded by the x-axis and the lines $y = 3 x , \quad x = 2$