Question

warm-up
students in a marine biology program built underwater drones. the students were split into two groups, and each group
designed and built its own drone. during a recent round of testing, they modeled the path of each drone using these
functions, where x is the horizontal distance from the launch site, in miles, and d(x) is the depth below the surface of the
water, in meters.
voyager: ( d(x) = 0.25(x - 35)^2 - 225 )
pioneer: ( d(x) = 0.3(x - 32)^2 - 270 )
use the custom button to graph the functions and determine the depth of each drone at its lowest point.
graph and buttons omitted

Explanation:

Step1: Identify Function Form

Both functions are in vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. For a parabola with \( a > 0 \), the vertex is the minimum point.

Step2: Find Voyager's Lowest Depth

For Voyager: \( d(x) = 0.25(x - 35)^2 - 225 \). Here, \( k = -225 \), so the minimum depth is \(-225\) meters (depth below surface, so magnitude is 225 m).

Step3: Find Pioneer's Lowest Depth

For Pioneer: \( d(x) = 0.3(x - 32)^2 - 270 \). Here, \( k = -270 \), so the minimum depth is \(-270\) meters (depth below surface, magnitude 270 m).

Answer:

Voyager's lowest depth: \(\boldsymbol{-225}\) meters (or 225 meters below surface).
Pioneer's lowest depth: \(\boldsymbol{-270}\) meters (or 270 meters below surface).