| № |
Condition |
free/or 0.5$ |
| m52601 | A manufacturer of corrugated metal roofing wants to produce panels that are 28 in. wide and 2 in. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has equation y = sin (π x/7) and find the width w of a flat metal sheet that is needed to make a 28-inch panel. (Use your calculator to evaluate the integral correct to four significant digits.) |
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| m52602 | A manufacturer of light bulbs wants to produce bulbs that last about 700 hours but, of course, some bulbs burn out faster than others. Let F(t) be the fraction of the company s bulbs that burn out before hours, so F(t) always lies between 0 and 1.
(a) Make a rough sketch of what you think the graph of F might look like.
(b) What is the meaning of the derivative r(t) = F(t)?
(c) What is the value of
Why? |
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| m52604 | A manufacturing company owns a major piece of equipment that depreciates at the (continuous) rate f = f(t) , where is the time measured in months since its last overhaul. Because a fixed cost A is incurred each time the machine is overhauled, the company wants to determine the optimal time T (in months) between overhauls.
(a) Explain
Why represents the loss in value of the machine over the period of time t since the last overhaul.
(b) Let C = C(t) be given by
What does represent and why would the company want to minimize C?
(c) Show that C has a minimum value at the numbers t = T Where C(T) = f(T) |
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| m52610 | A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is x(t) = 8 sin t, where is in seconds and in centimeters.
(a) Find the velocity and acceleration at time t.
(b) Find the position, velocity, and acceleration of the mass at time t = 2π/3. In what direction is it moving at that time? |
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| m52611 | A metal plate was found submerged vertically in sea water, which has density 64lb/ft3. Measurements of the width of the plate were taken at the indicated depths. Use Simpson s Rule to estimate the force of the water against the plate. |
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| m52612 | A mining company estimates that the marginal cost of extracting tons of copper ore from a mine is 0.6 + 0.008x, measured in thousands of dollars per ton. Start-up costs are $100,000. What is the cost of extracting the first 50 tons of copper? What about the next 50 tons? |
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| m52613 | A model for the US average price of a pound of white sugar from 1993 to 2003 is given by the function
S(t) = - 0.00003237t5 + 0.0009037t4 - 0.008956t3
+ 0.03629t2 - 0.04458t + 0.4074
where is measured in years since August of 1993. Estimate the times when sugar was cheapest and most expensive during the period 1993-2003. |
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| m52618 | (a) Newton s Law of Gravitation states that two bodies with masses m1 and m2 attract each other with a force
F = G m1m2 / r2
Where r is the distance between the bodies and G is the gravitational constant. If one of the bodies is fixed, find the work needed to move the other from r = a to r = b.
(b) Compute the work required to launch a 1000-kg satellite vertically to a height of 1000 km. You may assume that the earth s mass is 5.98 × 1024 kg and is concentrated at its center. Take the radius of the earth to be 6.37 × 106 m and G = 6.67 × 10-11 N ∙ m2/kg2. |
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| m52625 | A particle moves according to a law of motion s = f(t), t ≥ 0, where is measured in seconds and in feet.
(a) Find the velocity at time t.
(b) What is the velocity after 3 s?
(c) When is the particle at rest?
(d) When is the particle moving in the positive direction?
(e) Find the total distance traveled during the first 8 s.
(f) Draw a diagram like Figure 2 to illustrate the motion of the particle.
(g) Find the acceleration at time and after 3 s.
(h) Graph the position, velocity, and acceleration functions for 0 ≤ t ≤ 8. |
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| m52626 | A particle moves along a straight line with displacement s(t) velocity v(t), and acceleration a(t). Show that
A(t) = v(t) dv/dx
Explain the difference between the meanings of the derivatives dv/dt and dv/dx. |
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| m52627 | A particle moves along a straight line with equation of motion s = f(t), where is measured in meters and in seconds. Find the velocity and the speed when t = 5.
F(t) = 100 + 50 t - 4.9 t2 |
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| m52628 | A particle moves on a straight line with velocity function v(t) = sin ωt cos2 ωt. . Find its position function s= f(t) if f(0) = 0 |
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| m52629 | A particle moves on a vertical line so that its coordinate at time is y = t3 - 12 t + 3, t ≥ 0.
(a) Find the velocity and acceleration functions.
(b) When is the particle moving upward and when is it moving downward?
(c) Find the distance that the particle travels in the time interval 0 ≤ t ≤ 3.
(d) Graph the position, velocity, and acceleration functions for 0 ≤ t ≤ 3.
(e) When is the particle speeding up? When is it slowing down? |
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| m52630 | A particle that moves along a straight line has velocity v(t) = t2e-t meters per second after seconds. How far will it travel during the first seconds? |
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| m52634 | A patient takes 150 mg of a drug at the same time every day. Just before each tablet is taken, 5% of the drug remains in the body.
(a) What quantity of the drug is in the body after the third tablet? After the nth tablet?
(b) What quantity of the drug remains in the body in the long run? |
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| m52635 | A peach pie is taken out of the oven at 5:00 PM. At that time it is piping hot, 1000C. At 5:10 PM its temperature is 800C; at 5:20 PM it is 650C. What is the temperature of the room? |
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| m52641 | A phase trajectory is shown for populations of rabbits (R) and foxes (F).
(a) Describe how each population changes as time goes by.
(b) Use your description to make a rough sketch of the graphs of R and F as functions of time. |
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| m52645 | A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is π/3, this angle is decreasing at a rate of π/6 rad/min. How fast is the plane traveling at that time? |
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| m52646 | A plane flying horizontally at an altitude of 1 mi and a speed of 500 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.
(a) What quantities are given in the problem?
(b) What is the unknown?
(c) Draw a picture of the situation for any time t.
(d) Write an equation that relates the quantities.
(e) Finish solving the problem. |
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| m52647 | A plane flying with a constant speed of 300 km/h passes over a ground radar station at an altitude of 1 km and climbs at an angle of 30 °. At what rate is the distance from the plane to the radar station increasing a minute later? |
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