Honors AP Calculus / Mr. Hansen
3/8/2011

Name: _________________________
Bonus (for Mr. Hansen’s use only): ________

Test on Chapters 10 and 11 (Calculator Required)

 

Rules
  • You may not write calculator notation anywhere unless you cross it out. For example, fnInt(X^2,X,1,2) is not allowed; write  instead.
  • Adequate justification is required for free-response questions.
  • All final answers in free-response portions should be circled or boxed.
  • Decimal approximations must be correct to at least 3 places after the decimal point.

 

 

1.

A tetrahedron, as you recall from geometry class, is a 4-sided space figure. Compute the mass of a tetrahedral solid whose base is an equilateral triangle with sides of length 4 cm, given that the altitude of the figure (i.e., perpendicular distance from apex to the center of the base) is 3 cm. It is also given that density varies linearly with height, decreasing from 5 g/cm3 in the plane of the base to 2 g/cm3 at the top of the figure. (See sketch on board.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.

Find

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.(a)

Carefully sketch the particle motion defined for time t (in seconds), , by the vector function  Make your sketch rather large, since you will be adding more details to it in part (b). Linear units are centimeters (cm).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)

Compute the acceleration vector, as well as its tangential and normal components, when t = 2.5 seconds. Sketch all 3 vectors on your sketch above.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c)

Compute the particle’s speed at t = 2.5 seconds.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(d)

Briefly state the geometric meaning of the correct answer to part (c), regardless of whether or not you were able to compute it.

4.

Sam Swimmer is treading water in an extremely large, calm rectangular lake, 30 m from shore (perpendicular distance), when he notices an attractive young woman 65 m to one side of the point on the shore that is closest to him. She is at the water’s edge, and neither she nor Sam is at all close to one of the lake’s corners. Sam can swim at 0.85 m/sec and can walk briskly at 2 m/sec. Make a sketch and find the optimal point toward which Sam should swim so as to reach the young woman as quickly as possible. (Assume, contrary to reality, that she is indifferent toward him, and she moves neither toward him nor away from him.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.

The dollar cost, per foot of roadway, of building a 2-lane bridge across a rocky chasm between two sheer cliffs can be estimated as a function of the height h above ground (which varies, of course, for different points of the roadway) and the distance d to the nearest side of the bridge. All length measurements are in feet for this problem. The bridge runs from −400 ft. to +400 ft. relative to a center point that is 450 ft. above ground at the origin. The height profile of ground (not the bridge) can be closely approximated by the function y(x) = 450 for  y(x) =  for | x | < 400. The cost function, in dollars per foot, is given by 3.5hd.

 

 

(a)

Sketch the ground profile for x between −500 and +500 feet, as well as the roadway profile. Use a dotted line for the roadway profile, which is level at +450 feet throughout.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)

What type of application of the calculus is illustrated by the cost function and the associated total cost of the bridge? ___________  ___________  ___________

 

 

(c)

Compute the total estimated cost of the bridge, to the nearest dollar.