# Webassign Answers

The following page provides answers to many Webassign questions (mainly Calculus 1). Just browse/search the list of questions and enter the required data.

A sample of a radioactive substance decayed to % of its original amount after a year. (Round your answers to two decimal places.)
What is the half-life of the substance?
yr
How long would it take the sample to decay to % of its original amount?
yr

A cone-shaped paper drinking cup is to be made to hold cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper. (Round your answers to two decimal places.)
height cm
radius cm

A plane flies horizontally at an altitude of km and passes directly over a tracking telescope on the ground. When the angle of elevation is `pi`/, this angle is decreasing at a rate of `pi`/ rad/min. How fast is the plane traveling at that time? (Round your answer to two decimal places.)
km/min

A water trough is m long and has a cross-section in the shape of an isosceles trapezoid that is cm wide at the bottom, cm wide at the top, and has height cm. If the trough is being filled with water at the rate of m3/min how fast is the water level rising when the water is cm deep?
m/min

A bacteria culture grows with constant relative growth rate. After 2 hours there are bacteria and after 8 hours the count is .
Find the initial population.
P(0) = bacteria
Find an expression for the population after t hours.
P(t) =
Find the number of cells after hours.
P() = bacteria
Find the rate of growth after hours.
P'() = bacteria/hour
When will the population reach 200,000?
t = hours

At noon, ship A is km west of ship B. Ship A is sailing south at km/h and ship B is sailing north at km/h. How fast is the distance between the ships changing at 4:00 PM? (Round your answer to one decimal place.)
km/h

Point masses of mass m1, m2, and m3 are placed at the points (, 0), (, 0), and (0, ). Suppose that m1 = .
Find m2 such that the center of mass lies on the y-axis.

The velocity of a car was read from its speedometer at 10-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car. (Use the Midpoint Rule with 5 subintervals. Round your answer to one decimal place.)
mi
t(s)
v(mi/h)
0
0
10
20
52
30
40
55
50
60
56
70
80
50
90
100
45

The radius of a circular disk is given as cm with a maximum error in measurement of 0.2 cm.
Use differentials to estimate the maximum error in the calculated area of the disk. (Round your answer to two decimal places.)
cm2
What is the relative error? (Round your answer to four decimal places.)
What is the percentage error? (Round your answer to two decimal places.)
%

Water flows from the bottom of a storage tank at a rate of r(t) = − t liters per minute, where 0 ≤ t ≤ 50. Find the amount of water that flows from the tank during the first minutes.
liters

If $ is borrowed at % interest, find the amounts due at the end of years if the interest is compounded as follows.
Annually dollars
Semiannually dollars
Quarterly dollars
Monthly dollars
Weekly dollars
Daily dollars
Hourly dollars
Continuously dollars

The point P(, 1) lies on the curve y = √ x - 8 . If Q is the point (x, √ x - 8 ), use a scientific calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x.

A company estimates that the marginal cost (in dollars per item) of producing x items is - x. If the cost of producing one item is $, find the cost of producing 100 items. (Round your answer to two decimal places.)
$

A lighthouse is located on a small island km away from the nearest point P on a straight shoreline and its light makes revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? (Round your answer to one decimal place.)
km/min

If g(x) = f(f(x)), use the table to estimate the value of g'(1).
x
f(x)
0
0.5
1
1.5
2
2.5

A farmer wants to fence an area of million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence?
y = - x2
ft (smaller value)
ft (larger value)

When a cold drink is taken from a refrigerator, its temperature is 5°C. After 25 minutes in a 20°C room its temperature has increased to 10°C. (Round your answers to two decimal places.)
What is the temperature of the drink after minutes?
°C
When will its temperature be °C?
min

How long will it take an investment to double in value if the interest rate is % compounded continuously?
It will take years

A box with a square base and open top must have a volume of cm3. Find the dimensions of the box that minimize the amount of material used.
sides of base cm
height cm

Suppose that f(2) = , g(2) = , f' (2) = , and g'(2) = . Find h'(2).
h(x)=f(x)-g(x)
h'(2) =
h(x)=f(x)g(x)
h'(2) =
h(x)=f(x)/g(x)
h'(2) =
h(x)=g(x)/(1+f(x))
h'(2) =

Find the length of the shadow of a man who is 6 feet tall and is standing 15 feet from a streetlight that is h= feet high (see figure). (Round your answer to two decimal places.)
ft

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. (Round your answer to the nearest whole number.)
If the temperature of the turkey is 150°F after half an hour, what is the temperature after minutes?
T() = °F
When will the turkey have cooled to °?
t = min

A trough is ft long and its ends have the shape of isosceles triangles that are ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of ft3/min, how fast is the water level rising when the water is inches deep?
ft/min

The half-life of cesium-137 is 30 years. Suppose we have a mg sample.
Find the mass that remains after t years.
y(t) = mg
How much of the sample remains after years? (Round your answer to two decimal places.)
y() = mg
After how long will only 1 mg remain? (Round your answer to one decimal place.)
t = years

A television camera is positioned ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let's assume the rocket rises vertically and its speed is ft/s when it has risen ft. (Round your answers to three decimal places.)
How fast is the distance from the television camera to the rocket changing at that moment?
ft/s
If the television camera is always kept aimed at the rocket, how fast is the camera's angle of elevation changing at that same moment?
rad/s

The number of yeast cells in a laboratory culture increases rapidly at first but levels off eventually. The population is modeled by the function below, where t is measured in hours. At time t = 0 the population is cells and is increasing at a rate of cells/hour.
`n=f(t)=a/(1+b e^(-0.6t))`
Find the values of a and b.
a =
b =
According to this model, at what number of cell does the yeast population stabilize in the long run?
cells.

A model used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) is y = kN/(+N2), where k is a positive constant. What nitrogen level gives the best yield?
N =

Scientist can determine the age of ancient objects by a method called radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, 14C, with a half-life of about 5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates 14C through food chains. When a plant or animal dies, it stops replacing its carbon and the amount of 14C begins to decrease through radioactive decay. Therefore, the level of radioactivity must also decay exponentially.
A parchment fragment was discovered that had about % as much 14C radioactivity as does plant material on Earth today. Estimate the age of the parchment. (Round your answer to the nearest hundred years.)
Age of parchment is yr

For what values of the numbers a and b does the function below have maximum value f(1) = ? (Round the answers to three decimal places.)
`f(x) = ax e^(bx^2)`
a =
b =

What constant acceleration is required to increase the speed of a car from mi/h to mi/h in s? (Round your answer to two decimal places.)
ft/s2

A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to .
Find an expression for the number of bacteria after t hours.
P(t) =
Find the number of bacteria after hours. (Round your answer to the nearest whole number.)
P() = bacteria
Find the rate of growth after 4 hours. (Round your answer to the nearest whole number.)
P'() = bacteria per hour
When will the population reach 10,000? (Round your answer to one decimal place.)
t = hr

A Ferris wheel with a radius of m is rotating at a rate of one revolution every minutes. How fast is a rider rising when the rider is m above ground level?
m/min

If F(x) = f(f(f(x))), where f(0) = 0 and f '(0) = , find F '(0).
F'(0) =

Water is leaking out of an inverted conical tank at a rate of cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank. (Round your answer to the nearest integer.)
cm3/min

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is m from the dock? (Round your answer to two decimal places.)
m/min

The radius of a sphere is increasing at a rate of mm/s. How fast is the volume increasing when the diameter is mm? Evaluate your answer numerically. (Round the answer to the nearest whole number.)
mm3/s

Find the equation of the line through the point (, ) that cuts off the least area from the first quadrant. Give your answer using the form below.
y − A = B(x − C)
A =
B =
C =

Suppose that a population of bacteria triples every hour and starts with bacteria. Find an expression for the number n of bacteria after t hours.
n(t) =
Use it to estimate the rate of growth of the bacteria population after hours. (Round your answer to the nearest whole number.)
n'() = bacteria/hr

A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building? (Round your answer to two decimal places.)
m/s

A large drain and a small drain are opened to drain a pool. The large drain can empty the pool in h. After both drains have been open for h., the large drain becomes clogged and is closed. The small drain remains open and requires more hours to empty the pool. How long would it have taken the small drain, working alone, to empty the pool?
hours

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is ft, find the value of x so that the greatest possible amount of light is admitted. (Give your answer correct to two decimal places.)
x = ft

Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola. (Round your answers to the nearest hundredth.)
y = - x2
units (width)
units (height)

If a snowball melts so that its surface area decreases at a rate of cm2/min, find the rate at which the diameter decreases when the diameter is cm. (Give your answer correct to 4 decimal places.)
cm/min

A poster is to have an area of in2 with inch margins at the bottom and sides and a inch margin at the top. What dimensions will give the largest printed area? (Give your answers correct to one decimal place.)
in (width)
in (height)

The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is $ per month. A market survey suggests that, on average, one additional unit will remain vacant for each $ increase in rent. What rent should the manager charge to maximize revenue?
$ per month

Use differentials to estimate the amount of paint needed to apply a coat of paint cm thick to a hemispherical dome with diameter m. (Round your answer to two decimal places.)
m3

The minute hand on a watch is mm long and the hour hand is mm long. How fast is the distance between the tips of the hands changing at one o'clock? (Round your answer to one decimal place.)
mm/h

A plane flying horizontally at an altitude of mi and a speed of 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 mi away from the station. (Round your answer to the nearest whole number.)
mi/h

Gravel is being dumped from a conveyor belt at a rate of ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is ft high? (Round your answer to two decimal places.)
ft/min

Find the point on the parabola y2=2x that is closest to the point (1, ). (You may round your answers to two decimal places.)
(, )

A plane flying with a constant speed of km/h passes over a ground radar station at an altitude of 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? (Round your answer to the nearest whole number.)
km/h

Find two numbers whose difference is and whose product is a minimum.
(smaller number)
(larger number)

A car is traveling at 50 mi/h when the brakes are fully applied, producing a constant deceleration of ft/s2. What is the distance covered before the car comes to a stop? (Round your answer to one decimal place.)
ft

A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of ft/s.
At what rate is his distance from second base decreasing when he is halfway to first base? (Round your answer to one decimal place.)
ft/s
At what rate is his distance from third base increasing at the same moment? (Round your answer to one decimal place.)
ft/s

Two people start from the same point. One walks east at mi/h and the other walks northeast at mi/h. How fast is the distance between the people changing after 15 minutes? (Round your answer to three decimal places.)
mi/h

Each side of a square is increasing at a rate of cm/s. At what rate is the area of the square increasing when the area of the square is cm2?
cm2/s

Bismuth-210 has a half-life of 5.0 days.
A sample originally has a mass of 800 mg. Find a formula for the mass remaining after t days.
y(t) = `800e^(-ln(2)/5 t)`
Find the mass remaining after days.
mg
When is the mass reduced to 1 mg?
days
Sketch the graph of the mass function. (Do this on paper. Your teacher may ask you to turn in this work.)

When we estimate distances from velocity data, it is sometimes necessary to use times t0, t1, t2, t3, . . . that are not equally spaced. We can still estimate distances using the time periods Δti = ti − ti− 1. For example, a space shuttle was launched on a mission, the purpose of which was to install a new motor in a satellite. The table provided gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. Use these data to estimate the height, h, above Earth's surface of the space shuttle, 62 seconds after liftoff. (Give the upper approximation available from the data.)
h = ft
Event
Time (s)
Velocity (ft/s)
Launch
0
Begin roll maneuver
10
End roll maneuver
15
Throttle to 89%
20
Throttle to 67%
32
Throttle to 104%
59
Maximum dynamic pressure
62
Solid rocket booster separation
125

Find the point on the line x + y = that is closest to the point (, ). (Give your answers correct to two decimal places.)
(, )

Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths cm and cm if two sides of the rectangle lie along the legs.
cm2

A car is traveling at km/h when the driver sees an accident m ahead and slams on the brakes. What minimum constant deceleration is required to stop the car in time to avoid a pileup? (Round your answer to two decimal places.)
m/s2

A rectangular storage container with an open top is to have a volume of 10 m3. The length of this base is twice the width. Material for the base costs $ per square meter. Material for the sides costs $ per square meter. Find the cost of materials for the cheapest such container. (Round your answer to the nearest cent.)
$

A high-speed bullet train accelerates and decelerates at the rate of ft/s2. Its maximum cruising speed is mi/h. (Round your answers to three decimal places.)
What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 15 minutes?
mi
Suppose that the train starts from rest and must come to a complete stop in 15 minutes. What is the maximum distance it can travel under these conditions?
mi
Find the minimum time that the train takes to travel between two consecutive stations that are miles apart.
min
The trip from one station to the next takes at minimum 37.5 minutes. How far apart are the stations?
mi

The length of a rectangle is increasing at a rate of cm/s and its width is increasing at a rate of cm/s. When the length is cm and the width is cm, how fast is the area of the rectangle increasing?
cm2/s

If two resistors with resistances `R_1` and `R_2` are connected in parallel, as in the figure below, then the total resistance `R`, measured in ohms (`Omega`), is given by `1/R=1/R_1+1/R_2`
If `R_1` and `R_2` are increasing at rates of 0.3 `Omega \/s` and 0.2 `Omega \/s`, respectively, how fast is `R` changing when `R_1` = `Omega` and `R_2` = `Omega`? (Round your answer to three decimal places.)
`Omega \/s`

Find a cubic function, in the form below, that has a local maximum value of at and a local minimum value of at .
`f(x)=ax^3+bx^2+cx+d`
`f(x)=`

A powerful earthquake had a magnitude of on the Richter scale. An earthquake several years earlier was times as intense. What was the magnitude of the earlier earthquake?

A machinist is required to manufacture a circular metal disk with area cm2. (Round your answers to four decimal places.)
What radius produces such a disk?
cm
If the machinist is allowed an error tolerance of ± cm2 in the area of the disk, how close to the ideal radius in part (a) must the machinist control the radius?
cm

Find the dimensions of a rectangle with perimeter m whose area is as large as possible.
m (smaller value)
m (larger larger)

A runner sprints around a circular track of radius m at a constant speed of 7 m/s. The runner's friend is standing at a distance m from the center of the track. How fast is the distance between the friends changing when the distance between them is m? (Round your answer to two decimal places.)
m/s

If x2 + y2 = and dy/dt = , find dx/dt when y = .
±

The table gives estimates of the world population, in millions, from 1750 to 2000. (Round your answers to the nearest million.)
Year
Population
1750
790
1800
980
1850
1260
1900
1650
1950
2560
2000
6080
Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900 and 1950.
1900 million people
1950 million people
Use the exponential model and the population figures for and to predict the world population in 1950.
million people
Use the exponential model and the population figures for 1900 and 1950 to predict the world population in 1950.
million people

Brain weight B as a function of body weight W in fish has been modeled by the power function B = 0.007W2/3, where B and W are measured in grams. A model for body weight as a function of body length L (measured in centimeters) is W = 0.12L2.53. If, over 10 million years, the average length of a certain species of fish evolved from cm to cm at a constant rate, how fast was this species' brain growing when the average length was cm? (Round your answer to four significant figures.)
g/yr

The table gives the population of the United States, in millions, for the years 1900-2000.
Year
Population
1900
76
1910
92
1920
106
1930
123
1940
131
1950
150
1960
179
1970
203
1980
227
1990
250
2000
275
Use the exponential model and the census figures for and to predict the population in 2000.
P(2000) =
Use the exponential model and the census figures for and to predict the population in 2000.
P(2000) =

There are Cub Scouts in Troop 645. The number of scouts is more than five times the number of adult leaders. Find the number of adult leaders.

A particle moves along the curve below.
`y=sqrt(1+x^3)`
As it reaches the point (, ), the y-coordinate is increasing at a rate of cm/s. How fast is the x-coordinate of the point changing at that instant?
cm/s

A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of ft/s along a straight path. How fast is the tip of his shadow moving when he is ft from the pole?
ft/s

If h(x) = √ + f(x) , where f() = and f '() = ,
find h'().

Suppose that f(5) = , f'(5) = , g(5) = , and g'(5) = . Find the following values.
(fg)'(5);
(f/g)'(5);
(g/f)'(5);

A freshly brewed cup of coffee has temperature 95°C in a 20°C room. When its temperature is °C, it is cooling at a rate of 1°C per minute. When does this occur? (Round your answer to two decimal places.)
min

A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute.
t (min)
Heartbeats
36
38
40
42
44
The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient's heart rate after 42 minutes using the secant line between the points with the given values of t. (Round your answers to one decimal place.)
t = 36 and t = 42;
t = 38 and t = 42;
t = 40 and t = 42;
t = 42 and t = 44;