Module 4 Lesson 5 Formative Assessment Q
1. A curve is defined by the equation $$(x+y)^2-3xy=9$$ Find the rate of change
of the slope of the curve at the point $$(0,3)$$ .
Make sure you are using implicit differentiation.
*$$\frac{1}{2}$$
$$-\frac{3}{2}$$
$$-\frac{3}{4}$$
$$-\frac{1}{4}$$
2. A balloon rises vertically at the rate of 10 ft/s. A lamppost stands 20 ft from the
spot below the balloon. The lamppost is 25 ft high. At what rate is the shadow of
the balloon moving when the balloon is 15 ft above the ground?
Draw a diagram to help you see what is happening in the situation. Be sure to
properly label all given information, write an equation, and use a derivative.
$$\frac{25}{2} \text{ ft/s}$$
$$8 \text{ ft/s}$$
$$25 \text{ ft/s}$$
*$$50 \text{ ft/s}$$
3. A ladder 15 ft in length leans against a vertical wall, with the bottom of the
ladder 5 ft from the wall on a horizontal floor. If at that time the bottom of the
ladder is being pulled away at the rate of 2 ft/s, at what rate does the top of the
ladder slip down the wall?
Draw a diagram to help you see what is happening in the situation. Be sure to
properly label all given information, write an equation, and use a derivative.
*$$-\frac{\sqrt{2}}{2} \text{ ft/s}$$
$$-\frac{2\sqrt{5}}{3} \text{ ft/s}$$
$$-1\text{ ft/s}$$
$$-3\text{ ft/s}$$
4. As a balloon in the shape of a sphere is being blown up, the volume is increasing
at the rate of 4 in3/sec. At what rate is the radius increasing when the radius is 1
in?
Draw a diagram to help you see what is happening in the situation. Be sure to
properly label all given information, write an equation, and use a derivative.
*$$\frac{1}{\pi} \text{ in/s}$$
$$\frac{1}{64\pi} \text{ in/s}$$
$$\frac{1}{16\pi} \text{ in/s}$$
$$\frac{1}{4\pi} \text{ in/s}$$
5. A machine is rolling a metal cylinder under pressure. The radius of the cylinder is
decreasing at a constant rate of 0.05 inches per second and the volume V is
$$128\pi$$ cubic inches. At what rate is the length h changing when the radius r is
2.5 inches?
Draw a diagram to help you see what is happening in the situation. Be sure to
properly label all given information, write an equation, and use a derivative.
20.48 in/s
-0.819 in/s
*0.819 in/s
-16.38 in/s
Module 4 Lesson 5 Summative Assessment Q
1. A satellite is moving in an elliptical orbit about a planet. The equation of its
planar orbit is $$3x^2y+4y^2=20$$ . If the velocity of the satellite in the y
direction is 10 when the y coordinate of the satellite is 2, what is the velocity in the
x direction at that time?
*$$\pm \frac{15\sqrt{6}}{2}$$
$$\pm \frac{4\sqrt{3}}{2}$$
$$\pm \frac{20\sqrt{6}}{3}$$
$$\pm \frac{2\sqrt{3}}{5}$$
2. A balloon rises vertically at the rate of 10 ft/s. A lamppost stands 20 ft from the
spot below the rising balloon. The lamppost is 25 ft high. On the other side of the
balloon stands a vertical wall, 10 ft from the balloon (30 ft from the lamppost). At
what rate is the shadow moving on the wall when the balloon is 15 ft above the
ground?
$$\frac{5}{2} \text{ ft/s}$$
$$\frac{40}{3} \text{ ft/s}$$
*$$15\text{ ft/s}$$
$$5\text{ ft/s}$$
3. A ladder leans against a vertical wall, with the bottom of the ladder 8 ft from the
wall on a horizontal floor. At that time the bottom end of the ladder is being pulled
away at the rate of 3 ft/s and the top of the ladder slips down the wall at the rate of
4 ft/s. How long is the ladder?
6 ft
8 ft
* 10 ft
15 ft
4. A spherical balloon is inflated at the rate of 4 ft3/min. What is the volume of the
balloon when the radius is increasing at the rate of 6 in/min?
*$$\frac{8}{3}\sqrt{\frac{2}{\pi}} \text{ ft^3}$$
$$\frac{3}{8}\sqrt{\frac{3}{\pi}} \text{ ft^3}$$
$$\frac{\sqrt{6\pi}}{6\pi} \text{ ft^3}$$
$$\frac{2\sqrt{6\pi}}{3\pi} \text{ ft^3}$$
5. A machine is rolling a metal cylinder under pressure. The radius of the cylinder is
decreasing at a constant rate of 0.05 inches per second and the volume V is
$$128\pi$$ cubic inches. At what rate is the length h changing when the radius r is
1.5 inches?
-75.85 in/s
56.89 in/s
*3.793 in/s
9.481 in/s
Related Rates Wiki
You will be assigned to a group to work out a problem. Work with your
group to provide a complete, detailed solution. You will then be assigned
to constructively critique another group’s work as well.
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1. Water is leaking out of an inverted conical tank at a rate of 10,000
cm3/min at the same time 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.
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2. A trough is 10 ft long and its ends have the shape of isosceles triangles
that are 3 ft across at the top and have a height of 1 ft. If the trough is
filled with water at a rate of 12 ft3/min, how fast does the water level rise
when the water is 6 in deep?
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3. A water trough is 10 m long and a cross-section has the shape of an
isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the
top, and has a height of 50 cm. If it is being filled with water at the rate of
0.2 m3/min, how fast is the water level rising when the water is 30 cm
deep?
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4. Boyle’s Law states that when a sample of gas is compressed at a
constant temperature, the pressure P and volume V satisfy the equation
PV=C, where C is a constant. Suppose that at a certain instant the volume
is 600 cm3, the pressure is 150 kPa, and the pressure is increasing at a
rate of 20 kPa/min. At what rate is the volume decreasing at this instant?
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5. When air expand adiabatically (without gaining or losing heat), its
pressure P and volume V are related by the equation PV1.4=C where C is a
constant. Suppose that at a certain instant the volume is 400 cm3 and the
pressure is 80 kPa and is decreasing at a rate of 10 kPa/min. At what rate
is the volume increasing at this instant?
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6. 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 degrees.
At what rate is the distance from the plane to the radar station increasing
1 minute later?
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7. A ladder 10 ft long rests against a vertical wall. If the bottom of the
ladder slides away from the wall at a speed of 2 ft/s, how fast is the angle
between the top of the ladder and the wall changing when the angle is
$$\frac{\pi}{4}$$ radians?
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8. A lighthouse is on a small island 3 km away from the nearest point P on
a straight shoreline and its light makes four revolutions per minute. How
fast is the beam of light moving along the shoreline when it is 1 km from
point P?
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9. 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 1.6 m/s, how
fast is his shadow on the building decreasing when he is 4 m from the
building?
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10. A baseball diamond is a square with side 90 ft. A batter hits the ball
and runs toward first base with a speed of 24 ft/s. At what rate is his
distance from second base decreasing when he is half way to first base?