Question

a skydiver jumps from a plane that is 1000 ft above the ground. the function ( y = -16t^2 + 1000 ) gives the jumper’s height ( y ) (in feet) during the jump.
a. how long is the jumper in free fall if the parachute opens at 1000 ft?
b. how long is the jumper in free fall if the parachute opens at 500 ft?
c. what is a reasonable domain and range for the function?
(round to two decimal places as needed)

1. what is a reasonable domain for the function?

a. ( -1000 leq t leq 1000 )
b. ( 0 leq t leq 1000 )
c. ( 0 leq t leq 10 )
d. ( -10 leq t leq 10 )
e. all real numbers
what is a reasonable range for the function?
a. ( 0 leq y leq 1000 )
b. ( -10 leq y leq 10 )
c. ( -1000 leq y leq 1000 )
d. ( 0 leq y leq 10 )
e. all real numbers

Explanation:

Domain:

The function models the skydiver's height over time. Time \( t \) cannot be negative (since we start timing when the jump occurs), and we need to find when the height \( y = -16t^2 + 1000 \) reaches 0 (when the skydiver lands). Solving \( -16t^2 + 1000 = 0 \) gives \( t^2=\frac{1000}{16}=62.5 \), so \( t=\sqrt{62.5}\approx7.91 \). So time \( t \) is between 0 and approximately 8 (or 10, but more accurately up to when they land). Option C (\( 0\leq t\leq10 \)) is reasonable as it includes the time from jump (t=0) until landing (t≈7.91) and a bit more for context, while other options like A (negative time) or B (up to 1000, which is too large) don't make sense.

Range:

The height \( y = -16t^2 + 1000 \) starts at \( t=0 \) with \( y=1000 \) (initial height) and decreases to \( y=0 \) (when landing). So the range should be from 0 (or slightly above, but practically 0) up to 1000. Option A (\( 0\leq y\leq1000 \)) fits because height can't be negative (realistically, from 0 to 1000 feet), while other options like B (range of t, not y) or C (negative y) are incorrect.

Answer:

Domain: C. \( 0\leq t\leq10 \)

Range: A. \( 0\leq y\leq1000 \)