A farmer plans to make a shelter for starting tomatoes using two sheets of plexiglass and...

 

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A farmer plans to make a shelter for starting tomatoes using two sheets of plexiglass and two wooden sticks, as shown. One sheet is 1 meter by 2 meters, the other sheet is 2 meters by 2 meters, and the sticks are both h meters long, as shown. How long should the sticks be (in centimeters) so that the shelter contains the greatest possible volume between the plexiglass and the ground? Note: This volume equals the area of the triangle ABC times the distance AD. Glue these edges B/ 2 2 A E C A ►B 1 C Here are some hints and steps you should follow to get full marks • Define your variables. I recommend that you use V = V(h) for the volume. Find and label important distances on a diagram. • Find an expression for V(h). You may want to break the triangle ABC in to two smaller triangles to do this. • What is the domain of V? • Show that V'(h) = 1-2h? 4-2h2 V4-h • When finding the critical point(s), it may help to observe that if X +Y = 0 then X? = (-Y)2. You may need some care and patience for this step. • It is probably easier to check the endpoints of the domain of V than to use the second derivative method. • Write your answer in a sentence. A farmer plans to make a shelter for starting tomatoes using two sheets of plexiglass and two wooden sticks, as shown. One sheet is 1 meter by 2 meters, the other sheet is 2 meters by 2 meters, and the sticks are both h meters long, as shown. How long should the sticks be (in centimeters) so that the shelter contains the greatest possible volume between the plexiglass and the ground? Note: This volume equals the area of the triangle ABC times the distance AD. Glue these edges B/ 2 2 A E C A ►B 1 C Here are some hints and steps you should follow to get full marks • Define your variables. I recommend that you use V = V(h) for the volume. Find and label important distances on a diagram. • Find an expression for V(h). You may want to break the triangle ABC in to two smaller triangles to do this. • What is the domain of V? • Show that V'(h) = 1-2h? 4-2h2 V4-h • When finding the critical point(s), it may help to observe that if X +Y = 0 then X? = (-Y)2. You may need some care and patience for this step. • It is probably easier to check the endpoints of the domain of V than to use the second derivative method. • Write your answer in a sentence.

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