Arch Profile and Maximum Height

The question provides an equation that describes the profile of an arch ordered by a political party for their annual convention entrance. The equation is given as \(y = 2x - 0.1x^2\), where \(y\) represents the height of the arch in meters. Our goal is to determine the maximum possible height of this arch.

This equation is a quadratic equation, which typically represents a parabola. To make it easier to recognize the coefficients, we can rewrite the equation in the standard form \(y = ax^2 + bx + c\):

• Given equation: \(y = 2x - 0.1x^2\)
• Rearranging: \(y = -0.1x^2 + 2x + 0\)

From this standard form, we can identify the coefficients:

• \(a = -0.1\)
• \(b = 2\)
• \(c = 0\)

Since the coefficient \(a\) (which is -0.1) is negative, the parabola opens downwards. A parabola that opens downwards has a maximum point, which is also known as its vertex. The height of this vertex will be the maximum possible height of the arch.

Maximum Height Calculation

To find the coordinates of the vertex of a parabola defined by \(y = ax^2 + bx + c\), we use the formula for the x-coordinate:

\(x_{vertex} = \frac{-b}{2a}\)

Let's substitute the values of \(a\) and \(b\) from our arch equation:

• \(x_{vertex} = \frac{-2}{2(-0.1)}\)
• \(x_{vertex} = \frac{-2}{-0.2}\)
• \(x_{vertex} = 10\)

This value of \(x = 10\) meters represents the horizontal position at which the arch reaches its maximum height. To find the maximum height itself, we need to substitute this \(x\) value back into the original equation for \(y\):

\(y_{max} = 2(10) - 0.1(10)^2\)

Now, we perform the calculation:

• First, calculate \(2(10)\): \(20\)
• Next, calculate \((10)^2\): \(100\)
• Then, calculate \(0.1(100)\): \(10\)
• Finally, substitute these values back into the equation:
• \(y_{max} = 20 - 10\)
• \(y_{max} = 10\)

Therefore, the maximum possible height of the arch is 10 meters.

Final Answer for Arch Height

Based on the calculations, the maximum height that the political party's arch can reach is 10 meters.