Question

select the correct answer from each drop - down menu. a cable on a suspension bridge can be modeled by this equation, where h is the cable’s height, in feet, above the roadway 9 feet below the entrance to the bridge. ( h = 0.01x^{2}-0.3x + 22.5 ) rewrite the equation in vertex form, and then use it to complete this statement. the (\boxed{quad}) height of the cable is (\boxed{quad}) feet, and it occurs (\boxed{quad}) feet from the edge of the bridge. reset next

Explanation:

The given equation is \( h = 0.01x^2 - 0.3x + 22.5 \), which is a quadratic equation in the form \( y = ax^2 + bx + c \). To find the vertex (which gives the minimum height since the coefficient of \( x^2 \) is positive, meaning the parabola opens upwards), we can use the formula for the x - coordinate of the vertex of a quadratic function \( x=-\frac{b}{2a} \), and then substitute this x - value back into the equation to find the corresponding y - value (the height).

Step 1: Identify \( a \), \( b \), and \( c \)

For the quadratic equation \( h = 0.01x^2 - 0.3x + 22.5 \), we have \( a = 0.01 \), \( b=- 0.3 \), and \( c = 22.5 \).

Step 2: Find the x - coordinate of the vertex

Using the formula \( x=-\frac{b}{2a} \), substitute \( a = 0.01 \) and \( b=-0.3 \):

$$ x=-\frac{- 0.3}{2\times0.01}=\frac{0.3}{0.02} = 15 $$

This x - value represents the distance from the edge of the bridge (in feet) where the minimum height occurs.

Step 3: Find the minimum height (y - coordinate of the vertex)

Substitute \( x = 15 \) back into the equation \( h = 0.01x^2 - 0.3x + 22.5 \):

$$ h=0.01\times(15)^2-0.3\times15 + 22.5 $$

First, calculate \( (15)^2=225 \), then \( 0.01\times225 = 2.25 \), and \( 0.3\times15=4.5 \)

$$ h=2.25-4.5 + 22.5=20.25 $$

Answer:

The minimum height of the cable is \( 20.25 \) feet, and it occurs \( 15 \) feet from the edge of the bridge.