The curve y = 0.005x^2 - 2x + 200 for 0 ≤ x ≤ 200 and the line segment from F(0, 200) to G(200, 0)

The curve y = 0.005x2 - 2x + 200 for 0 ≤ x ≤ 200 and the line segment from F(0, 200) to G(200, 0) are shown in the standard (x, y) coordinate plane below.

1. What is the y-coordinate for the point on the curve with x-coordinate 20 ?

  1. 160
  2. 162
  3. 164
  4. 166
  5. 168

2. The length of this curve is longer than FG. About how many coordinate units long is FG?

  1. 20
  2. 141
  3. 200
  4. 283
  5. 400

3. Tran wants to approximate the area underneath the curve y = 0.005x2 - 2x + 200 for 0 ≤ x ≤ 200, shown shaded in the graph below.

He finds an initial estimate, A, for the shaded area by using FG and computing

A = 1/2(200 units)(200 units) = 20,000 square units.

The area of the shaded region is:

  1. less than 20,000 square units, because the curve lies under FG.
  2. less than 20,000 square units, because the curve lies over FG.
  3. equal to 20,000 square units.
  4. greater than 20,000 square units, because the curve lies under FG.
  5. greater than 20,000 square units, because the curve lies over FG.

Answer

1. The correct answer is B.

If the x-coordinate is 20, then the y-coordinate can be found by substituting 20 for x.

y = 0.005(20)2 - 2(20) + 200

= 0.005(400) - 40 + 200

= 0.5(4) + 160

= 2 + 160 = 162

2. The correct answer is D.

The distance formula or the Pythagorean theorem gives this distance directly.

FG = √(2002 + 2002) = 200√2

= 283 (approximately)

3. The correct answer is A.

The shaded region is entirely contained in the given triangle because the curve is below the hypotenuse of the triangle, FG. The area of the triangle is made up of the shaded area plus the unshaded area above the curve and inside the triangle.

So, the shaded area is less than the area of the triangle.