Question

alberta diploma question
a rain gutter is made from sheets of aluminum that are 25 cm wide. the first step in forming
the rain gutter is to turn the edges up to form right angles, as shown in the diagram below

the cross - sectional area formed by the turned - up edges affects the water flow. this cross -
sectional area, a, can be modelled by the function
$x(25 - 2x)$
where $x$ is the height of the turned - up edges.

to the nearest tenth of a centimetre, the height of the turned - up edge, $x$, that will
maximize the cross - sectional area is
cm

Explanation:

Step1: Identify the function type

The cross - sectional area function is \( A(x)=x(25 - 2x)=- 2x^{2}+25x \), which is a quadratic function in the form \( y = ax^{2}+bx + c \) where \( a=-2 \), \( b = 25 \), and \( c = 0 \).

Step2: Find the vertex of the quadratic function

For a quadratic function \( y=ax^{2}+bx + c \), the x - coordinate of the vertex (which gives the maximum or minimum value of the function) is given by the formula \( x=-\frac{b}{2a} \).
Substitute \( a=-2 \) and \( b = 25 \) into the formula:
\( x=-\frac{25}{2\times(-2)}=\frac{25}{4}=6.25 \)
Rounding to the nearest tenth, we get \( x = 6.3 \) (since the hundredth digit is 5, we round up the tenth digit).

Answer:

\( 6.3 \)