An arch is built so that it has the shape of a parabola with the equation `y=-3x^(2)+24x` where y represents the height of the arch in meters. How many times greater is the maximum height of the arch than the width of the arch at its base?

Options

A. `4`
B. `6`
C. `8`
D. `10`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the equation of the parabola
The given equation of the arch is:
\[ y = -3x^2 + 24x \]

### Step 2: Find the points where the arch cuts the x-axis
To find the width of the arch at its base, we need to determine where the arch intersects the x-axis. This occurs when \( y = 0 \):
\[ -3x^2 + 24x = 0 \]
Factoring out the common term:
\[ -3x(x - 8) = 0 \]
Setting each factor to zero gives:
\[ x = 0 \quad \text{or} \quad x = 8 \]

### Step 3: Calculate the width of the arch at its base
The width of the arch at its base is the distance between the two x-intercepts:
\[ \text{Width} = 8 - 0 = 8 \text{ meters} \]

### Step 4: Find the maximum height of the arch
The maximum height of the arch occurs at the vertex of the parabola. The x-coordinate of the vertex for a parabola in the form \( y = ax^2 + bx + c \) is given by:
\[ x = -\frac{b}{2a} \]
Here, \( a = -3 \) and \( b = 24 \):
\[ x = -\frac{24}{2 \times -3} = \frac{24}{6} = 4 \]

### Step 5: Calculate the maximum height by substituting \( x = 4 \) back into the equation
Now we substitute \( x = 4 \) into the original equation to find the maximum height:
\[ y = -3(4^2) + 24(4) \]
Calculating this:
\[ y = -3(16) + 96 = -48 + 96 = 48 \text{ meters} \]

### Step 6: Calculate how many times greater the maximum height is than the width
Now, we need to find how many times greater the maximum height is than the width:
\[ \text{Ratio} = \frac{\text{Maximum Height}}{\text{Width}} = \frac{48}{8} = 6 \]

### Final Answer
The maximum height of the arch is 6 times greater than the width of the arch at its base.

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