| № |
Condition |
free/or 0.5$ |
| m52351 | a.
b.
c. |
buy |
| m52352 | A bacteria culture contains 200 cells initially and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells.
(a) Find the number of bacteria after hours.
(b) Find the number of bacteria after 4 hours.
(c) Find the rate of growth after 4 hours.
(d) When will the population reach 10,000? |
buy |
| m52353 | A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420.
(a) Find an expression for the number of bacteria after hours.
(b) Find the number of bacteria after 3 hours.
(c) Find the rate of growth after 3 hours.
(d) When will the population reach 10,000? |
buy |
| m52354 | A bacteria population is 4000 at time t = 0 and its rate of growth is 1000 . 2 bacteria per hour after t hours. What is the population after one hour? |
buy |
| m52357 | A balloon is rising at a constant speed of 5 ft/s. A boy is cycling along a straight road at a speed of 15 ft/s. When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 s later? |
buy |
| m52364 | A Bernoulli differential equation (named after James Bernoulli) is of the form
dy / dx + P(x) y = Q(x)yn
Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, show that the substitution u = y1-n transforms the Bernoulli equation into the linear equation
dy / dx + 1 (1 - n) P(x) u = (1 - n) = Q(x) |
buy |
| m52365 | A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x. |
buy |
| m52366 | (a) By graphing the function f(x) = (cos 2x - cos x) / x2 and zooming in toward the point where the graph crosses the -axis, estimate the value of limx→0 f(x).
(b) Check your answer in part (a) by evaluating f(x) for values of x that approach 0. |
buy |
| m52367 | (a) By reading values from the given graph of f, use four rectangles to find a lower estimate and an upper estimate for the area under the given graph of f from x = 0 to x = 8. In each case sketch the rectangles that you use.
(b) Find new estimates using eight rectangles in each case. |
buy |
| m52368 | A cable with linear density p = 2 kg/m is strung from the tops of two poles that are 200 m apart.
(a) Use Exercise 52 to find the tension so that the cable is 60 m above the ground at its lowest point. How tall are the poles?
(b) If the tension is doubled, what is the new low point of the cable? How tall are the poles now? |
buy |
| m52375 | A car is moving with speed 20 m/s and acceleration 2 m/s2 at a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate the distance traveled during the next minute? |
buy |
| m52379 | A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square centimeters, are 0, 18, 58, 79, 94, 106, 117, 128, 63, 39, and 0. Use the Midpoint Rule to estimate the volume of the liver. |
buy |
| m52380 | (a) Cavalieri s Principle states that if a family of parallel planes gives equal cross-sectional areas for two solids S1 and S2, then the volumes of S1 and S2 are equal. Prove this principle.
(b) Use Cavalieri s Principle to find the volume of the oblique cylinder shown in the figure. |
buy |
| m52381 | A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maxi - mum brightness is 5.4 days. The average brightness of this star is 4.0 and its brightness changes by ± 0.35. In view of these data, the brightness of Delta Cephei at time t, where t is measured in days, has been modeled by the function
B(t) = 4.0 + 0.35 sin (2πt/5.4)
(a) Find the rate of change of the brightness after days.
(b) Find, correct to two decimal places, the rate of increase after one day. |
buy |
| m52399 | A clepsydra, or water clock, is a glass container with a small hole in the bottom through which water can flow. The "clock" is calibrated for measuring time by placing markings on the container corresponding to water levels at equally spaced times. Let x = f(x) be continuous on the interval [0, b] and assume that the container is formed by rotating the graph of f about the y-axis. Let V denote the volume of water and the height of the water level at time t.
(a) Determine V as a function of h.
(b) Show that
dV/dt = π[f(h)]2 dh/dt
(c) Suppose that A is the area of the hole in the bottom of the container. It follows from Torricelli s Law that the rate of change of the volume of the water is given by
dV/dt = kA √h
Where is a negative constant. Determine a formula for the function f such that dh/dt is a constant C. What is the advantage in having dh/dt = C? |
buy |
| m52409 | A company modeled the demand curve for its product ( in dollars) by the equation
P = (800000e-x/5000)/(x+20,000)
Use a graph to estimate the sales level when the selling price is $16. Then find (approximately) the consumer surplus for this sales level. |
buy |
| m52431 | A container in the shape of an inverted cone has height 16 cm and radius 5 cm at the top. It is partially filled with a liquid that oozes through the sides at a rate proportional to the area of the container that is in contact with the liquid. (The surface area of a cone is πr/, where r is the radius and is the slant height.) If we pour the liquid into the container at a rate of, then the height of the liquid decreases at a rate of 0.3 cm3/min when the height is 10 cm. If our goal is to keep the liquid at a constant height of 10 cm, at what rate should we pour the liquid into the container? |
buy |
| m52435 | A cross-section of an airplane wing is shown. Measurements of the thickness of the wing, in centimeters, at 20-centimeter intervals are, 5.8, 20.3, 26.7, 29.0, 27.6, 27.3, 23.8, 20.5, 15.1, 8.7, and 2.8. Use the Midpoint Rule to estimate the area of the wing s cross-section. |
buy |
| m52436 | A cube with 20-cm-long sides is sitting on the bottom of an aquarium in which the water is one meter deep. Estimate the hydrostatic force on
(a) The top of the cube
(b) One of the sides of the cube. |
buy |
| m52437 | A curve, called a witch of Maria Agnesi, consists of all possible positions of the point P in the figure. Show that parametric equations for this curve can be written as
x = 2a cot θ, y = 2a sin2θ
Sketch the curve. |
buy |