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Math Problems
Calculus
Find the rate of change of one variable when rate of change of other variable is given
The area of a square is increasing at a rate of
2
2
2
square inches per second. At the time when the area of the square is
1
1
1
, what is the rate of change of the perimeter of the square? Round your answer to three decimal places (if necessary).
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The temperature in a room changes at a rate of
r
(
t
)
=
ln
(
t
+
1
)
sin
(
t
)
r(t)=\ln (t+1) \sin (t)
r
(
t
)
=
ln
(
t
+
1
)
sin
(
t
)
degrees Celsius per hour (where
t
t
t
is the time in hours). At
t
=
1
t=1
t
=
1
hour, the temperature is
18
18
18
degrees Celsius.
\newline
What is the room's temperature at time
t
=
3
t=3
t
=
3
hours?
\newline
Use a graphing calculator and round your answer to three decimal places.
\newline
degrees Celsius
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Snow is piling on a driveway so its depth is changing at a rate of
r
(
t
)
=
10
1
−
cos
(
0.5
t
)
r(t)=10 \sqrt{1-\cos (0.5 t)}
r
(
t
)
=
10
1
−
cos
(
0.5
t
)
centimeters per hour, where
t
t
t
is the time in hours,
0
≤
t
≤
5
0 \leq t \leq 5
0
≤
t
≤
5
. At time
t
=
0
t=0
t
=
0
, the depth of the snow is
20
20
20
centimeters.
\newline
What is the snow's depth at time
t
=
5
t=5
t
=
5
hours?
\newline
Use a graphing calculator and round your answer to three decimal places.
\newline
centimeters
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The radius of the base of a cone is decreasing at a rate of
2
2
2
centimeters per minute.
\newline
The height of the cone is fixed at
9
9
9
centimeters.
\newline
At a certain instant, the radius is
13
13
13
centimeters.
\newline
What is the rate of change of the volume of the cone at that instant (in cubic centimeters per minute)?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
78
π
-78 \pi
−
78
π
\newline
(B)
−
156
π
-156 \pi
−
156
π
\newline
(C)
−
12
π
-12 \pi
−
12
π
\newline
(D)
−
507
π
-507 \pi
−
507
π
\newline
The volume of a cone with radius
r
r
r
and height
h
h
h
is
π
r
2
h
3
\pi r^{2} \frac{h}{3}
π
r
2
3
h
.
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The altitude of an aeroplane is
500
500
500
metres, and the angle of elevation from the runway to the aeroplane is
1
5
∘
15^{\circ}
1
5
∘
. Find the horizontal distance from the aeroplane to the runway, to the nearest centimetre.
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The vertical distance from the dock to the boat's mast reaches its highest value of
−
27
c
m
-27 \mathrm{~cm}
−
27
cm
every
3
3
3
seconds. The first time it reaches its highest point is after
1
1
1
.
3
3
3
seconds. Its lowest value is
−
44
c
m
-44 \mathrm{~cm}
−
44
cm
.
\newline
Find the formula of the trigonometric function that models the vertical height
H
H
H
between the dock and the boat's mast
t
t
t
seconds after Antonio starts his stopwatch. Define the function using radians.
\newline
H
(
t
)
=
□
H(t)=\square
H
(
t
)
=
□
\newline
What is the vertical distance
2
2
2
.
5
5
5
seconds after Antonio starts his stopwatch? Round your answer, if necessary, to two decimal places.
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Water is drained out of a tank at a rate of
r
(
t
)
=
20
e
−
0.1
t
2
r(t)=20 e^{-0.1 t^{2}}
r
(
t
)
=
20
e
−
0.1
t
2
liters per minute, where
t
t
t
is the time in minutes,
0
≤
t
≤
10
0 \leq t \leq 10
0
≤
t
≤
10
.
\newline
How much water is drained between times
t
=
3
t=3
t
=
3
and
t
=
9
t=9
t
=
9
minutes?
\newline
Use a graphing calculator and round your answer to three decimal places.
\newline
liters
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The radius of a sphere is decreasing at a rate of
1
1
1
meter per hour.
\newline
At a certain instant, the radius is
4
4
4
meters.
\newline
What is the rate of change of the volume of the sphere at that instant (in cubic meters per hour)?
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The volume of a rectangular prism is
150
c
m
3
150 \mathrm{~cm}^{3}
150
cm
3
. Alex measures the sides to be
9.89
c
m
9.89 \mathrm{~cm}
9.89
cm
by
3.43
c
m
3.43 \mathrm{~cm}
3.43
cm
by
5.08
c
m
5.08 \mathrm{~cm}
5.08
cm
. In calculating the volume, what is the relative error, to the nearest hundredth.
\newline
Answer:
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The volume of a rectangular prism is
80
c
m
3
80 \mathrm{~cm}^{3}
80
cm
3
. Alex measures the sides to be
4.53
c
m
4.53 \mathrm{~cm}
4.53
cm
by
2.02
c
m
2.02 \mathrm{~cm}
2.02
cm
by
7.69
c
m
7.69 \mathrm{~cm}
7.69
cm
. In calculating the volume, what is the relative error, to the nearest hundredth.
\newline
Answer:
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The volume of a rectangular prism is
144
f
t
3
144 \mathrm{ft}^{3}
144
ft
3
. Alex measures the sides to be
2.63
f
t
2.63 \mathrm{ft}
2.63
ft
by
7.54
f
t
7.54 \mathrm{ft}
7.54
ft
by
6.13
f
t
6.13 \mathrm{ft}
6.13
ft
. In calculating the volume, what is the relative error, to the nearest thousandth.
\newline
Answer:
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The volume of a rectangular prism is
150
i
n
3
150 \mathrm{in}^{3}
150
in
3
. Alex measures the sides to be
2
2
2
.
8
8
8
in by
9
9
9
.
72
72
72
in by
4
4
4
.
54
54
54
in. In calculating the volume, what is the relative error, to the nearest hundredth.
\newline
Answer:
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The volume of a rectangular prism is
504
i
n
3
504 \mathrm{in}^{3}
504
in
3
. Eric measures the sides to be
6
6
6
.
63
63
63
in by
7
7
7
.
76
76
76
in by
9
9
9
.
08
08
08
in. In calculating the volume, what is the relative error, to the nearest hundredth.
\newline
Answer:
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The volume of a rectangular prism is
360
c
m
3
360 \mathrm{~cm}^{3}
360
cm
3
. Alex measures the sides to be
9.09
c
m
9.09 \mathrm{~cm}
9.09
cm
by
4.84
c
m
4.84 \mathrm{~cm}
4.84
cm
by
8.35
c
m
8.35 \mathrm{~cm}
8.35
cm
. In calculating the volume, what is the relative error, to the nearest hundredth.
\newline
Answer:
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The volume of a rectangular prism is
108
f
t
3
108 \mathrm{ft}^{3}
108
ft
3
. David measures the sides to be
6.14
f
t
6.14 \mathrm{ft}
6.14
ft
by
3.44
f
t
3.44 \mathrm{ft}
3.44
ft
by
5.8
f
t
5.8 \mathrm{ft}
5.8
ft
. In calculating the volume, what is the relative error, to the nearest hundredth.
\newline
Answer:
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The volume of a rectangular prism is
54
f
t
3
54 \mathrm{ft}^{3}
54
ft
3
. Carter measures the sides to be
2.85
f
t
2.85 \mathrm{ft}
2.85
ft
by
9.32
f
t
9.32 \mathrm{ft}
9.32
ft
by
2.09
f
t
2.09 \mathrm{ft}
2.09
ft
. In calculating the volume, what is the relative error, to the nearest thousandth.
\newline
Answer:
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A cylinder has a base radius of
6
6
6
feet and a height of
8
8
8
feet. What is its volume in cubic feet, to the nearest tenths place?
\newline
Answer:
V
=
V=
V
=
feet
3
{ }^{3}
3
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A cylinder has a base diameter of
20
f
t
20 \mathrm{ft}
20
ft
and a height of
10
f
t
10 \mathrm{ft}
10
ft
. What is its volume in cubic
f
t
\mathrm{ft}
ft
, to the nearest tenths place?
\newline
Answer:
V
=
V=
V
=
f
t
3
\mathrm{ft}^{3}
ft
3
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A cylinder has a base radius of
5
c
m
5 \mathrm{~cm}
5
cm
and a height of
16
c
m
16 \mathrm{~cm}
16
cm
. What is its volume in cubic
c
m
\mathrm{cm}
cm
, to the nearest tenths place?
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What is the volume of a hemisphere with a diameter of
52.5
f
t
52.5 \mathrm{ft}
52.5
ft
, rounded to the nearest tenth of a cubic foot?
\newline
Answer:
f
t
3
\mathrm{ft}^{3}
ft
3
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