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Calculus
Suppose that for a company manufacturing calculators, the cost, and revenue equations are given by
C=70000+30x, R=200−(x^2/20)
C=70000+30x,R=200−x220,
where the production output in one week is x calculators. If the production rate is increasing at a rate of 500 calculators when the production output is 6000 calculators, find each of the following:
Rate of change in cost =
Rate of change in revenue =
Rate of change in profit =
C=70000+30x, R=200−(x^2/20)
C=70000+30x,R=200−x220,
where the production output in one week is x calculators. If the production rate is increasing at a rate of 500 calculators when the production output is 6000 calculators, find each of the following:
Rate of change in cost =
Rate of change in revenue =
Rate of change in profit =
Calculus
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Suppose you have a street light at a height H. You drop a rock vertically so that it hits the ground at a distance d from the street light. Denote the height of the rock by h. The shadow of the rock moves along the ground. Let s denote the distance of the shadow from the point where the rock impacts the ground. Of course, s and h are both functions of time. use the notation v to denote h′: v=h′.
Then the speed of the shadow at any time while the rock is in the air is given by s′=........................ (where s′ is an expression depending on h, s, H, and v (You will find that d drops out of your calculation.) Now consider the time at which the rock hits the ground. At that time
h=s=0.
The speed of the shadow at that time is s′=.....................where your answer is an expression depending on H, v, and d.
Hint: Use similar triangles and implicit differentiation. For the second part of the problem you will need to compute a limit.
Suppose you have a street light at a height H. You drop a rock vertically so that it hits the ground at a distance d from the street light. Denote the height of the rock by h. The shadow of the rock moves along the ground. Let s denote the distance of the shadow from the point where the rock impacts the ground. Of course, s and h are both functions of time. use the notation v to denote h′: v=h′.
Then the speed of the shadow at any time while the rock is in the air is given by s′=........................ (where s′ is an expression depending on h, s, H, and v (You will find that d drops out of your calculation.) Now consider the time at which the rock hits the ground. At that time
h=s=0.
The speed of the shadow at that time is s′=.....................where your answer is an expression depending on H, v, and d.
Hint: Use similar triangles and implicit differentiation. For the second part of the problem you will need to compute a limit.
Calculus
Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 10 feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by V=1/3 πr^2h.
Calculus
Water is leaking out of an inverted conical tank at a rate of 10,000cm^3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6m and the diameter at the top is 4m. If the water level is rising at a rate of 20cm/min when the height of the water is 2m, find the rate at which water is being pumped into the tank.
=................cm^3/min
=................cm^3/min
Calculus
A particle is moving along the curve y=x√. As the particle passes through the point (4,2), its x-coordinate increases at a rate of 3cm/s. How fast is the distance from the particle to the origin changing at this instant?
................cm/s
................cm/s
Calculus
A boat is pulled into a dock by a rope attached to the bow ("front") of the boat and passing through a pulley on the dock that is 1m higher than the bow of the boat. If the rope is pulled in at a rate of 1m/s, how fast is the boat approaching the dock when it is 8m from the dock?
=..........m/s
=..........m/s
Calculus
Brain weight B as a function of body weight W in fish has been modeled by the power function B=.007W^(2/3), where B and W are measured in grams. A model for body weight as a function of body length L (measured in cm) is W=.12L^(2.53). If, over 10 million years, the average length of a certain species of fish evolved from 15cm to 20cm at a constant rate, how fast was the species' brain growing when the average length was 18cm? Round your answer to the nearest hundredth.
=......................nanograms/yr
=......................nanograms/yr
Calculus
A spotlight on the ground is shining on a wall 20m away. If a woman 2m tall walks from the spotlight toward the building at a speed of 0.8m/s, how fast is the length of her shadow on the building decreasing when she is 6m from the building?
Answer (in meters per second):
Answer (in meters per second):
Calculus
If z^2=x^2+y^2 with z>0, dx/dt=2, and dy/dt=5, find dz/dt when x=5 and y=12.
Answer: dz/dt=
Answer: dz/dt=
Calculus
Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 3m/s, how fast is the area of the spill increasing when the radius is 70m?
Answer (in m^2/s):
Answer (in m^2/s):
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#340153 on Dec 2023
#340153 on Dec 2023