Question

Bernard is playing a game where he is tossing
2
bean bags underhand and trying to make them land at certain distances. He is standing in the center of a circular target with two rings painted on the ground.



Ring 1 starts at a horizontal distance of
5
feet from him and ends at a distance of
10
feet from him.
Ring 2 starts at a horizontal distance of
10
feet from him and ends at a distance of
15
feet from him.


image


The equations represent the height,
y
, in feet of each of Bernard's tosses as a function of
x
, the horizontal distance, in feet, the bean bag has traveled from Bernard.



First toss:
y
=
−
2
x
2
+
8
x
+
3

Second toss:
y
=
−
1
6
x
2
+
2
x
+
3



Which TWO statements are TRUE?



A.
Bernard will win the prize.

B.
Bernard will not win the prize.

C.
Bernard earned
5
points with the first toss.

D.
Bernard earned
10
points with the first toss.

E.
Bernard earned
5
points with the second toss.

F.
Bernard earned
10
points with the second toss.

Answer 1

Find where each bean bag hits ground (y = 0).

First toss: -2x^2 + 8x + 3 = 0
2x^2 - 8x - 3 = 0 → x = (4 ± √22)/2 ≈ 4.345 or -0.345. Positive landing distance ≈ 4.35 ft (< 5), so it misses both rings (0 points).

Second toss: -(1/6)x^2 + 2x + 3 = 0
x^2 - 12x - 18 = 0 → x = 6 ± 3√6 ≈ 13.348 or -1.348. Positive landing distance ≈ 13.35 ft (between 10 and 15), so it lands in Ring 2 → 10 points.

Total = 10 points, so he does not win the prize.

True statements: B and F.