Exam 8: Further Applications of Integration

According to the National Health Survey, the heights of adult males in the United States are (normally distributed with mean) 73 inches, and standard deviation of $2.8$ inches. What is the probability that an adult male chosen at random is between 65 inches and 75 inches tall? Select the correct answer.

The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2 minutes. The manager wants to advertise that anybody who isn't served within a certain number of minutes gets a free hamburger. But she doesn't want to give away free hamburgers to more than $5 \%$ of her customers. What value of $x$ must she use in the advertisement "if you aren't served within $x$ minutes, you get a free hamburger"?

Use Simpson's Rule with $n = 6$ to estimate the length of the curve $y = 3 \sin x , 0 \leq x \leq \pi$ . Round your answer to six decimal places.

The demand function for a commodity is given by $p = 2,000 - 0.1 x - 0.01 x ^ { 2 } .$ Find the consumer surplus when the sales level is 100 .

A movie theater has been charging $\$ 10.00$ per person and selling about 500 tickets on a typical weeknight. After surveying their customers, the theater estimates that for every $\$ 0.5$ that they lower the price, the number of moviegoers will increase by 50 per night. Find the demand function and calculate the consumer surplus when the tickets are priced at $\$ 8$ .

Write an integral giving the area of the surface obtained by revolving the curve about the $x$ -axis. (Do not evaluate the integral.) $y = \frac { 2 } { x } \text { on } [ 1,5 ]$

A large tank is designed with ends in the shape of the region between the curves $y = \frac { x ^ { 2 } } { 2 }$ and $y = 15$ , measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of $9 \mathrm { ft }$ with gasoline. (Assume that the density of the gasoline is $42.0 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .) Select the correct answer.

Dye dilution is a method of measuring cardiac output. If $A \mathrm { mg }$ of dye is used and $c ( t )$ is the concentration of the dye at time $t$ , then the cardiac output over the time interval $[ 0 , T ]$ is given by $F = \frac { A } { \int _ { 0 } ^ { T } c ( t ) d t }$ Find the cardiac output over the time interval $[ 0,15 ]$ if the dye dilution method is used with $11 \mathrm { mg }$ of dye and the dye concentration, in $\mathrm { mg } / \mathrm { L }$ , is modeled by $c ( t ) = \frac { 1 } { 2 } t ( 15 - t ) , 0 \leq t \leq 15$ where $t$ is measured in seconds.

Write an integral giving the area of the surface obtained by revolving the curve about the $x$ -axis. (Do not evaluate the integral.) $y = \frac { 4 } { x } \text { on } [ 3,6 ]$

A rectangular tank has width $3 \mathrm { ft }$ , height $2 \mathrm { ft }$ , and length $7 \mathrm { ft }$ . It is filled with equal volumes of water and oil. The oil has a weight density of $50 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ and floats on the water. Find the force exerted by the mixture on one end of the tank. (The weight density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .)

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve $y = \ln x ^ { 5 }$ about the $x$ -axis on the interval $1 \leq x \leq 5$ .

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve $y = \ln x ^ { 7 }$ about the $x$ -axis on the interval $1 \leq x \leq 7$ .

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .)

A hot, wet summer is causing a mosquito population explosion in a lake resort area. The number of mosquitoes is increasing at an estimated rate of $2,100 + 7 e ^ { 0.7 t }$ per week (where $t$ is measured in weeks). By how much does the mosquito population increase between the 4 th and 7 th weeks of summer?

The marginal revenue from producing $x$ units of a certain product is $100 + x - 0.001 x ^ { 2 } + 0.00003 x ^ { 3 }$ (in dollars per unit). Find the increase in revenue if the production level is raised from 1,100 units to 1,700 units. Select the correct answer.

A vertical plate is submerged in water (the surface of the water coincides with the $x$ -axis). Find the force exerted by the water on the plate. (The weight density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .)

Use Simpson's Rule with $n = 6$ to estimate the length of the curve $y = 3 \sin x , 0 \leq x \leq \pi$ . Round your answer to six decimal places. Select the correct answer.

A gate in an irrigation canal is constructed in the form of a trapezoid $4 \mathrm { ft }$ wide at the bottom, $8 \mathrm { ft }$ wide at the top, and $2 \mathrm { ft }$ high. It is placed vertically in the canal, with the water extending to its top. Find the hydrostatic force on one side of the gate.

The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2 minutes. Find the probability that a customer is served within the first 2 minutes. Select the correct answer.

A trough has vertical ends that are equilateral triangles with sides of length $7 \mathrm { ft }$ . If the trough is filled with water to a depth of $4 \mathrm { ft }$ , find the force exerted by the water on one end of the trough. Round to one decimal place. (The weight density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .) Select the correct answer.