Please show all your work. You have been hired by Ups & Downs, Inc. to help with

Question

Please show all your work.

You have been hired by Ups & Downs, Inc. to help with the design of a new roller coaster. A sketch of the desired path design of the proposed coaster is given below (straight stretch, no turns). A preliminary analysis shows that the height of the track given by the function H(x) in meters by where A and T are constants that will satisfy the Ups & Downs requirements. Construct your own roller coaster design by taking T to be ten plus the product of the last two digits of your student ID number. Determine the constant A by requiring the track lo end at ground level, exactly 175 meters from the start. It is ok if your track goes below ground (i.e., the x-axis); any portion (hat is below ground will be assumed lo be a straight path along the ground. Determine the following for your report: Where is the path increasing and decreasing? (Give your answer in terms of distance, in meters, along the ground from the start.) Locate the tops and the bottoms of the path.

Explanation / Answer

First plug all the numbers in, and you get

H(x) = 31/2 sin(x/10) + Ax + (31/2)(x-87.5)^2/ (87.5)^2 - 31/2

At 175 m,

H(175) = -15.1222 + A(175)

Solving for A you get

A = 0.086413 (I don't know how many decimal places you need, so i'll just leave a lot. Round off as needed)

Then the equation becomes

H(x) = 31/2 sin(x/10) + 0.086413x + (31/2)(x-87.5)^2/ (87.5)^2 - 31/2

To answer 1., derive

H'(x) = 31/20 cos(x/10) + 0.086413 + (31)(x-87.5)/(87.5)^2

Answering part 2 first makes more sense.

2.

Use graphing calculator to find max and min points

max: 14.350451, 78.871998, 143.40345

min: 47.608638, 108.8385, 170.0395

Max are the tops, mins are the bottoms

1. Initial section is increasing, and each section alternates between increasing and decreasing.

Increase: 0 to 14. 350451, 47.608638 to 78.871998, 108.8385 to 143.40345, 170.0395 to 175

Decrease: 14.350451 to 47.608638, 78.871998 to 108.8385, 143.40345 to 170.0395

3. Use same steps to find mins and maxes of H'(x) graph.

max: 0.26125362, 63.093107, 125.92496

min: 31.15467, 93.986521, 156.81838

Rate of change is increasing between min to max, and decreasing between max to min

Increasing rate: 0 to 0.26125362, 31.15467 to 63.093107, 93.986521 to 125.92496, 156.81838 to 175

Decreasing rate: 0.26125362 to 31.15467, 63.093107 to 93.986521, 125.92496 to 156.81838

Now just match various sections to see where things are doing what

Increasing at increasing rate: 0 to 0.26125362, 47.608638 to 63.093107, 108.8385 to 125.92496

Increasing at decreasing rate: 0.26125362 to 14.350451, 63.093107 to 78.871998, 125.92496 to 143.40345

Decreasing at increasing rate: 31.15467 to 47.608638, 93.986521 to 108.8385, 156.81838 to 175

Decreasing at decreasing rate: 14.350451 to 31.15467, 78.871998 to 93.986521, 143.40345 to 156.81838

4.

The zeros of the graph are

0, 27.550456, 68.905332, 88.968824, 126.60893, 164.83521, 175

Positive regions are

0 to 27.550456, 68.905332 to 88.968824, 126.60893 to 164.83521

The beam locations in these area is

10, 20, 70, 80, 130, 140, 150, 160

Calculate value of H(x) at each of these points

10.566518, 9.5464435, 1.3521735, 6.8619375, 5.9029603, 17.532176, 15.449513, 4.5047373

Square each of these values

111.6513026, 91.1345835, 1.828373174, 47.08618625, 34.8449403, 307.3771953, 238.6874519, 20.29265814

Add all these values together

852.9026912

5.

When calculating angle, I will assume that question means the average angle of the decent.

Note that the track does not go underground, so you must only think from top to zero point.

There are three tops.

Each decent is from a top to nearest zero to the right.

First decent is from 14.350451 to 27.550456

Height at top is 11.930213

Angle can be calculated by

angle = arctan(change in height / length of decent)

= arctan(11.930213 / (27.550456-14.350451))

= 0.7349126083

Second decent is from 78.871998 to 88.968824

Height at top is 6.9576905

Angle can be calculated by

angle = arctan(change in height / length of decent)

=arctan(6.9576905 / (88.968824 - 78.871995))

= 0.6033706993

Third decent is from 143.40345 to 164.83521

Height at top is 18.399959

Angle can be calculated by

angle = arctan(change in height / length of decent)

=arctan(18.399959 / (164.83521 - 143.40345))

= 0.7094294028

Using these numbers, total thrill of this coaster is

5.04771271

Again, round all value as needed.

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