One standard size for the United States flag is 19 feet by 10 feet. On a flag of this size, the union (the blue rectangle in the top-left corner) is 7 5 8 feet by 5 3 8 feet. Astronauts on the Apollo missions had a flag on a shoulder patch, use the scale factor 1/60 and find the union dimensions for the patch

Δabc is dilated by a scale factor of 3 with the origin as the center of dilation to form δa′b′c′. the slope of is -1.2. the length of is p units, the length of is q units, and the length of is r units. the slope of is . the length of is units.

Billy plotted −3 4 and −1 4 on a number line to determine that −3 4 is smaller than −1 4 is he correct? explain why or why not?

Prove (a) cosh2(x) − sinh2(x) = 1 and (b) 1 − tanh 2(x) = sech 2(x). solution (a) cosh2(x) − sinh2(x) = ex + e−x 2 2 − 2 = e2x + 2 + e−2x 4 − = 4 = . (b) we start with the identity proved in part (a): cosh2(x) − sinh2(x) = 1. if we divide both sides by cosh2(x), we get 1 − sinh2(x) cosh2(x) = 1 or 1 − tanh 2(x) = .