# Math Help - Comparing Quadratics

Please help my solve this question: Two rockets are shot into the air from the same location. The paths of the rockets are given by
#1: y = -0.025x^2 + 3x - 52.5,
and #2: y = -0.04x^2 + 0.1x + 40.
Which rocket goes higher, and by how much?

I'm uncertain as to how I should solve this. Assistance would be much appreciated. Thank you.

2. Hello, currypuff!

Two rockets are shot into the air from the same location.

The paths of the rockets are given by: . $\begin{array}{ccc} y_1 &=& -0.025x^2 + 3x - 52.5 \\ y_2 &=& -0.04x^2 + 0.1x + 40\end{array}$

Which rocket goes higher, and by how much?

Both graphs are down-opening parabolas; their maximums occur at their vertex.

. . Vertex formula: . $x \:=\:\frac{\text{-}b}{2a}$

For $y_1\!:\;a = -0.025,\;b = 3\quad\Rightarrow\quad x \:=\:\frac{-3}{2(-0.025)} \:=\:60$

Hence: . $y_1 \;=\;-0.025(60^2) + 3(60) - 52.5 \;=\;\boxed{37.5}$

For $y_2\!:\;a = 0.04,\;b = 0.1\quad\Rightarrow\quad x \:=\:\frac{-0.1}{2(0.04)} \:=\:1.25$

Hence: . $y_2\;=\;-0.04(1.25^2) + 0.1(1.25) + 40 \;=\;\boxed{40.0625}$

The second rocket goes 2.5625 units higher.