2011 FAMAT State Relay Round 0 Theta Let A equal the sum of the reciprocals of the zeros of 2 8 15 0x x− + = . Alpha The probability that a random ball in a hat is red is A. The probability that the ball is

blue is A2

. Let B equal the probability that a ball is neither red nor blue (assuming only

8A

balls are in the hat).

Calculus

C = 10

sinV

B

x dxπ

−∫ , where 252BV π

= .

Round 1 Theta

A ball is dropped from a height of 1800 feet and on every bounce returns 13

of its

previous height. Let A equal the total vertical distance the ball travels in feet. Alpha

( )sin 75B A= +o o Calculus

Let C equal dydx

for 2 24 3x y y+ = − at the point where ( )6 2x B= − and 1y ≠ − .

Round 2 Theta A equals the area of a regular hexagon with apothem length 3 3 (no units, please). Alpha

Consider the hyperbola 2 2 2A9 4 18 16 65 03

x y x y ⎛ ⎞− + − − − =⎜ ⎟

⎝ ⎠. Let B equal the x-

coordinate of the focus with largest x-coordinate. Calc Let C equal the total area bounded by (not just the integral) the graph of

( )sin 13y B x⎡ ⎤= −⎣ ⎦

and the x-axis on the interval [ ]0, 101π .

Round 3 Theta

2 2

3 3

1220

x yx yx y A

+ =

+ =

+ =

Hint: Consider ( )( )2 2x y x y+ + . Your answer should be negative. Alpha

sin672AB π⎛ ⎞= ⎜ ⎟

⎝ ⎠ (remember to rationalize)

Calculus Let C equal the volume of the solid obtained when the region bounded by the graph of

3 2sin ,2 4

By x xπ π−= ≤ ≤ and the x-axis is revolved around the x-axis.

Round 4 Calculus Let A equal the 2,667th derivative (with respect to x) of 2 sinxe x− evaluated at 0x = . Theta

2668 2670 2669

1 1 12 2 2A A AB − − −

= + +

Alpha

( )2 4 34logf x x x x= + + . Let 32

7BC f ⎛ ⎞= ⎜ ⎟

⎝ ⎠.

Round 5 Calculus

Calculate 1

116

sin (2 )x

d xdx

−

=

, multiply it by 7 , and let this equal A.

Theta

B equals the area of the ellipse 2 2

2 1164

x yA

+ =

Alpha

C equals the magnitude of the cross-product of 22i j kBπ

+ −r r r

and i j k+ +r r r

.

Round 6 Calculus Consider the region in the xy-plane bounded by the graph of 2 1, 1 1y x x= − + − ≤ ≤ and the x-axis. Let A equal the volume of the solid formed when cross-sections are taken as squares with bases in the region perpendicular to the x-axis. Theta

B equals the determinant of the inverse of 15 122 1

A⎡ ⎤−⎢ ⎥⎢ ⎥⎣ ⎦

.

Alpha

( )04 n

nC B

∞

=

=∑

Round 7 Alpha A equals the distance between the polar points ( )4, 30o and ( )6, 210o . Calculus B equals the slope of the tangent line to the graph of ( )2( ) lnf x x= at Ax e= . Theta

C equals ( )8

ln ln20eB

⎛ ⎞+ ⎜ ⎟

⎝ ⎠ expressed as a constant.

Round 8 Alpha A equals the area of the triangle (in units squared) with side lengths 6, 6, and 8 units. Calculus B equals the rate (in units/minute) the surface area of a sphere is changing the instant the

volume is 2415πA cubic units, and the radius is increasing by 1 units

min.π.

Theta C equals the distance between (2, 20) and (6, B).

Round 9 Alpha Given that 2x = is a double solution of 4 3 26 14 16 8 0x x x x− + − + = , if the other two roots are u and v , let A u v= + . Calculus

B equals 3

0

sin cosV

x x dx∫ , where 3 2AV π

= .

Theta

B is a fraction in simplest form. Let’s say uBv

= . Then let 3v

u v uC i i i i= + + + .