A model space shuttle is propelled into the air and is described by the equation y=-x^2/2e + ex (in 1000 ft), where y is its height above the ground. What is the maximum height that the shuttle reaches?
To find the maximum height that the space shuttle reaches, we need to determine the vertex of the parabolic equation y = -x^2/2e + ex. The vertex of a parabola is the highest or lowest point on the curve.
The equation is in the form y = ax^2 + bx, where a = -1/2e and b = e. The vertex of a parabola in this form is given by the coordinates x = -b/2a and y = f(x), where f(x) is the value of y at the vertex.
In this case, a = -1/2e and b = e. Substituting these values into the formula, we have:
x = -e / (2*(-1/2e))
= -e / (-1/e)
= -e^2
Now we can find y by substituting this value of x into the equation:
y = -(-e^2)^2 / (2e) + e(-e^2)
= e^4 / (2e) - e^3
= (e^4 - 2e^4) / (2e)
= - e^4 / (2e)
= - e^3 / 2
Therefore, the maximum height that the shuttle reaches is - e^3 / 2 (in 1000 ft).