Question

18 mark for review
a water tank is leaking water from a crack in its base. the amount of water, in hundreds of gallons, remaining in the tank t hours after the crack formed can be modeled by w, a decreasing function of time t. which of the following gives a verbal representation of the function ( w^{-1} ), the inverse function of w ?
a ( w^{-1} ) is an increasing function of the amount of time after the crack formed.
b ( w^{-1} ) is a decreasing function of the amount of time after the crack formed.
c ( w^{-1} ) is an increasing function of the amount of water in the tank.
d ( w^{-1} ) is a decreasing function of the amount of water in the tank.

Explanation:

The original function \( W(t) \) maps time \( t \) (hours) to remaining water \( W \) (hundreds of gallons) and is decreasing: as \( t \) increases, \( W \) decreases. The inverse function \( W^{-1}(w) \) swaps inputs and outputs, so it maps remaining water \( w \) to the time \( t \) when that amount of water is left. Since \( W(t) \) is decreasing, smaller \( w \) corresponds to larger \( t \), meaning as \( w \) increases, \( t \) decreases. Thus, \( W^{-1} \) is a decreasing function of the amount of water in the tank.

Answer:

D. \( W^{-1} \) is a decreasing function of the amount of water in the tank.