• Feb 8th 2008, 11:17 AM
currypuff
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.
• Feb 8th 2008, 11:53 AM
Soroban
Hello, currypuff!

Quote:

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

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

Which rocket goes higher, and by how much?

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

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

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

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

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

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

The second rocket goes 2.5625 units higher.