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Sample Problem, #2
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Given Problem. (#2 in text) |
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First we find a relationship between A and r. In this case we can use the area formula for a circle. |
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Now we differentiate both sides of the equation with respect to t. This is the answer to part (a) of the problem. |
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These are the given facts in the problem. (radius increases at a constant rate of 1 m/s, and radius = 30) |
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Here we substitute the given rate that the radius changes and the value of r and we can calculate the answer desired in part (b). |
Sample Problem, #10
| A spotlight 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 the length of his shadow on the building decreasing when he is 4 m from the building? | Given Problem. (#10 in text) |
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Here is my diagram for this problem.
S stands for the length of the Shadow on the wall. So using the letters
I have chosen, we will be looking for  . Notice
also that I am calling the distance of the person to the wall x. |
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We recognize that there are two similar right triangles in the diagram. This ratio of proportional sides says: "The bottom side of the small triangle is to the bottom side of the large triangle as the altitude of the small triangle is the the altitude of the large triangle. |
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Here we cross multiply the terms of the above proportion. Now we have an equation relating the two key items in the problem, namely S (shadow length) and x (distance of person from wall) |
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We differentiate both sides with respect to t. We needed to use the product rule on the left side. |
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We algebraically solve this equation
for . |
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Using the proportion from above, we can calculate S when x = 4. This will be needed for substitution in the next step. |
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So the speed that the length of the shadow is decreasing is .6 m/s |
Sample Problem, #30
| A lighthouse is located 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 P? | Given Problem, (#30 in textbook). |
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Here is a diagram which might represent the problem. y
represents the length of the shadow along the shoreline. We have been given We are to find |
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This important equation relates the
given information variable ( ) with
the variable that we are to find the rate of. (y) |
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Here we differentiate both sides of
the equation with respect to t.
Note that to take the derivative of the inverse tangent we used the
formula:  and we must also use the
chain rule twice. (once on the 
function and once on the y.) |
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Algebraic simplification. |
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Solve this equation for  . |
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We now have the rate of change of y with respect to t when y = 1. |
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1-27, Odds
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