| № |
Condition |
free/or 0.5$ |
| m52648 | A planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum.
The ore resembles pink talcum powder and is poured from a conveyor at the top of the silo.
The silo is a cylinder 100 ft high with a radius of 200 ft. The conveyor carries ore at a rate of 60,000π ft3/h and the ore maintains a conical shape whose radius is 1.5 times its height.
(a) If, at a certain time t, the pile is 60 ft high, how long will it take for the pile to reach the top of the silo?
(b) Management wants to know how much room will be left in the floor area of the silo when the pile is 60 ft high. How fast is the floor area of the pile growing at that height?
(c) Suppose a loader starts removing the ore at the rate of 20,000π ft3/h when the height of the pile reaches 90 ft. Suppose, also, that the pile continues to maintain its shape. How long will it take for the pile to reach the top of the silo under these conditions? |
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| m52651 | (a) Plot the point with polar coordinates (4, 2(/3). Then find its Cartesian coordinates?
(b) The Cartesian coordinates of a point are (-3, 3). Find two sets of polar coordinates for the point? |
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| m52653 | A population is modeled by the differential equation
dP/dt = 1.2P(1 - P/4200)
(a) For what values of P is the population increasing?
(b) For what values of P is the population decreasing? |
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| m52654 | A population of honeybees increased at a rate of r(t) bees per week, where the graph of r is as shown. Use the Midpoint Rule with six subintervals to estimate the increase in the bee population during the first 24 weeks. |
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| m52655 | A population of protozoa develops with a constant relative growth rate of 0.7944 per member per day. On day zero the population consists of two members. Find the population size after six days. |
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| m52661 | (a) Program a calculator or computer to use Euler s method to compute y(1) where y(x)is the solution of the initial-value problem
dy/dx + 3x2y = 6x2 y(0) = 3
(i) h = 1
(ii) h = 0.1
(iii) h = 0.01
(iv) h = 0.001
(b) Verify that y = 2 + e-x3 is is the exact solution of the differential equation.
(c) Find the errors in using Euler s method to compute y(1) with the step sizes in part (a). What happens to the error when the step size is divided by 10? |
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| m52663 | (a) Prove that the equation has at least one real root.
(b) Use your calculator to find an interval of length 0.01 that contains a root.
cos x = x3 |
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| m52664 | (a) Prove that the equation has at least one real root.
(b) Use your graphing device to find the root correct to three decimal places.
100e-x/100 = 0.01 x2 |
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| m52680 | A rechargeable battery is plugged into a charger. The graph shows C(t), the percentage of full capacity that the battery reaches as a function of time elapsed (in hours).
(a) What is the meaning of the derivative C (t)?
(b) Sketch the graph of C (t). What does the graph tell you? |
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| m52681 | A rectangle has perimeter 20 m. Express the area of the rectangle as a function of the length of one of its sides.
Find a formula for the described function and state its domain. |
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| m52694 | A roast turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 75°F.
(a) If the temperature of the turkey is 150°F after half an hour, what is the temperature after 45 minutes?
(b) When will the turkey have cooled to 100°F? |
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| m52696 | A runner sprints around a circular track of radius 100 m at a constant speed of 7 m/s. The runner s friend is standing at a distance 200 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200 m? |
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| m52706 | A sequence is defined recursively by the equations a1 = 1, an + 1 = 1/3 (an + 4). Show that {an} is increasing and an < 2. Show that {an} is increasing and for all. Deduce that is convergent and find its limit? |
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| m52709 | (a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the -axis and (ii) the -axis.
(b) Use the numerical integration capability of your calculator to evaluate the surface areas correct to four decimal places.
1. y = tan x, 0 ≤ x ≤ π/3
2. y = (e -x)2, - ≤ x ≤ 1 |
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| m52710 | (a) Set up an integral for the volume of a solid torus (the donut-shaped solid shown in the figure) with radii r and R.
(b) By interpreting the integral as an area, find the volume of the torus. |
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| m52713 | A ship is moving at a speed of 30 km/h parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon.
(a) Express the distance between the lighthouse and the ship as a function of , the distance the ship has traveled since noon; that is, find f so that s = f(d).
(b) Express as a function of , the time elapsed since noon; that is, find so that d = g (t).
(c) Find f o g. What does this function represent? |
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| m52714 | (a) Show that a polynomial of degree 3 has at most three real roots.
(b) Show that a polynomial of degree has at most real roots. |
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| m52716 | (a) Show that any function of the form
Y = A sinh mx + B cosh mx
satisfies the differential equation y" = m2y.
(b) Find y = y(x) such that y" = 9y, y(0) = -4, and y (0) = 6. |
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| m52717 | (a) Show that for 1 ≤ √(1+x3) ≤ 1 + x3 for x ≥ 0.
(b) Show that |
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| m52719 | (a) Show that for xy ( - 1.
arctan x - arctan y = arctan x - y/1 + xy
If the left side lies between - (/2 and = (/4.
(b) Show that arctan 120/119 - arctan 1/239 = (/4.
(c) Deduce the following formula of John Machin (1680 - 1751):
4 arctan 1/5 - arctan 1/239 = (/4
(d) Use the Maclaurin series for arctan to show that
0.1973955597 < arctan 1/5 < 0.1973955616
(e) Show that
0.004184075 < arctan 1/239 < 0.04184077
(f) Deduce that, correct to seven decimal places, ( ( 3.1415927.
Machin used this method in 1706 to find correct to 100 decimal places. Recently, with the aid of computers, the value of has been computed to increasingly greater accuracy. In 2009 T. Daisuke and his team computed the value of to more than two trillion decimal places! |
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