Exercises

Mount Suluclac, which is really just a hill, may be defined by the function y=h(x) = -0.0001x⁴ + 0.019x³ - 1.17x² + 27x for $x\in[0,100]$ . Here x is the horizontal distance in metres from the western base of the hill and y is the height of the hill also measured in metres. An orchard grows to the west of Mount Sulucalc, spanning the region from x = -60 to x=0.

Once h(x) is defined, enter the following lines in Maple V to get a picture of the hill and the orchard.

   > p1 := plot(h,0..100,color='brown',thickness=2):
   > p2 := plot(0,-60..0,color='khaki',thickness=5):
   > display([p1,p2]);

You will investigate how much sun and shade the orchard and Mount Suluclac receive on an average sunny day. Use the following simplified description of the movement of the sun.

The sun travels in a circular orbit centered at the origin, and passes directly overhead. At 6 a.m. the sun is positioned on the positive x-axis ( $\theta=0$ ), and at 6 p.m. the sun is on the negative x-axis ( $\theta = \pi$ ). Here $\theta$ is defined as the angle between the line connecting the origin to the sun and the positive x-axis. Since the sun is at a considerable distance from the earth and Mount Suluclac, its light arrives as parallel rays at angle $\theta$ to the positive x-axis.

Enter the following lines in Maple V to see the hill, the orchard and three rays of sunshine at 3 o'clock in the afternoon.

   > p3 := plot(-x,x=-70..-10,color='tan',thickness=2):
   > p4 := plot(-x+200,x=-70..10,color='tan',thickness=2):
   > p5 := plot(-x+400,x=-70..120,color='tan',thickness=2):
   > display([p1,p2,p3,p4,p5]);

Answer the questions below, discussing fully the strategy and procedure you use in obtaining the solutions.

1.

i.: What is the formula giving $\theta$ as a function of t, where t is the number of hours past midnight?
ii.: What is the slope of rays that come to earth at angle $\theta$ ?
iii.: What is the slope of the sun's rays t hours past midnight?

2.

i.: Find the coordinates of the point on the hillside that determines the edge of the shadow in the fruit orchard at 8am.
ii.: Where in the orchard is it sunny, and where in the orchard is it shady? Give the coordinate of the point in the orchard that represents the shady part from the sunny part.
iii.: Create a plot showing the hill, the orchard, and the sun's ray separating the regions of sunlight and shade.
iv.: Repeat this exercise for the times 10am, 11:30am, and 4pm.

3.

i.: Find the coordinates of any points on the hillside that separate sunlit and shady regions of the hillside at 8am.
ii.: Where on the hillside is it sunny, and where is it shaded?
iii.: Give a plot of the hill and the crucial sun's rays that separate the regions of sunlight and shade.
iv.: Repeat this exercise for the times 10am, 11:30am, and 4pm.

4.

There is a prizewinning strawberry at x=-27, within the fruit orchard. The following Maple V commands draw the hillside, the orchard, and the strawberries.

 > s1:=plot([[-27,0],[-26,0]], color='red', thickness=6):
 > display([p1,p2,s1]);

(The command draws strawberries from x=-27 to x=-26 so that they will show up on the plot. However, you should treat the strawberry patch as a single point in the following exercise.)

To obtain superior strawberries, the patch can receive no more than a certain number of hours of sunlight each day. Assuming that the patch has been positioned at x=-27 so that the strawberries receive the maximum recommended daily allowance of sun, find this recommended amount.

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Christine M Palmer
9/29/1998