Calc I word problems help

mfidai2

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Mar 4, 2015
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1) A space shuttle launches with altitude function a(t) = 20t2 (t = seconds and a(t) = meters). An observer standing 4 miles away (horizontally) from the launch pad must look up at a higher and higher angle as time goes by in order to watch the space shuttle. Calculate the rate of change of that angle after 1 second of lift-off.

2) Suppose that a traffic drone is 1000 feet in the air, directly over a straight freeway.
It observes a car on the freeway that is 2000 feet away from the drone traveling
away from the drone at 80 feet per second. The speed limit is 65 miles per hour.
Is the car breaking the speed limit? Justify your answer.


3) A coffee filter has the shape of an inverted cone. Water drains out of the filter at
a rate of 10cm3 per minute. When the depth of the water in the cone is 8cm, the
depth is decreasing at a rate of 2cm per minute. What is the ratio of the height of
the cone to its radius?
 
What are your thoughts? What have you tried? How far have you gotten? Where are you stuck?

1) A space shuttle launches with altitude function a(t) = 20t2 (t = seconds and a(t) = meters). An observer standing 4 miles away (horizontally) from the launch pad must look up at a higher and higher angle as time goes by in order to watch the space shuttle. Calculate the rate of change of that angle after 1 second of lift-off.
You drew the horizontal line for the ground (being the fixed distance between the viewer and the launch site), the vertical line for the launch path, and the slanty line between the shuttle and the viewer (being the viewer's line-of-sight). You noted that this was a right triangle, and that you are being asked about the rate of change of the base angle at the viewer's position, so you labelled this angle. (All of this, you learned back in algebra and trigonometry.)

You labelled the appropriate side of the triangle with the listed function. You noted that the ground distance was fixed, so dx/dt = 0. You found a way of relating the specified information to the angle. And... then what?

2) Suppose that a traffic drone is 1000 feet in the air, directly over a straight freeway. It observes a car on the freeway that is 2000 feet away from the drone traveling away from the drone at 80 feet per second. The speed limit is 65 miles per hour. Is the car breaking the speed limit? Justify your answer.
You drew the horizontal line for the freeway. You drew a vertical line from the drone's position down to the freeway. You drew a slanty line from the drone to the car's position. You labelled the vertical line with the given information, noting that this length is fixed, so dy/dt = 0. And... then what?

3) A coffee filter has the shape of an inverted cone. Water drains out of the filter at a rate of 10cm3 per minute. When the depth of the water in the cone is 8cm, the depth is decreasing at a rate of 2cm per minute. What is the ratio of the height of the cone to its radius?
You drew an upside-down triangle to represent the side view of the cone. You drew the height line from the vertex at the base, up through the middle of the triangle to the horizontal line at the top. You drew an horizontal line somewhere below the top of the triangle and the base angle, representing the decreasing depth line. You noted that you had right triangles and similar triangles, and that dh/dt = -2. You noted the formula for the volume of a cone with radius r (being half the "width" of the horizontal line for the water's height) and height h. And... then what?

Please be complete. Thank you! :wink:
 
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