Question

1. a civil engineering company is hired to design a new drainage system. to construct one of the storm drains, a sheet of metal that is 18 feet wide is folded on both sides. what is the height of the drain with the maximum cross - sectional area? a. 0 feet b. 4.5 feet c. 9 feet d. 18 feet 5. the function f(x)=(x + 5)(x - 3) is dilated by x to produce the function g(x)=x·f(x). how many x - intercepts does the function g(x) have? a. 0 b. 1 c. 2 d. 3 6. analyze the graph h(x). if f(x)=x² and g(x)=3x - 2, which of the following is true? a. h(x)=f(x)+g(x) b. h(x)=f(x)-g(x) c. h(x)=f(x)·g(x) d. h(x)=f(x)÷g(x) 7. what value can you dilate the function f(x)=x² by to produce a function with more than one x - intercept? a. 2 b. x c. x³ d. x - 2

Explanation:

4.

Step1: Let the height of the drain be $x$ feet.

The width of the base of the drain after folding on both sides is $18 - 2x$ feet. The cross - sectional area $A(x)$ of the drain (assuming a rectangular cross - section) is given by the product of the height and the base width, so $A(x)=x(18 - 2x)=18x-2x^{2}$.

Step2: Find the maximum of the quadratic function.

For a quadratic function $y = ax^{2}+bx + c$ (in our case, $a=-2$, $b = 18$, $c = 0$), the vertex of the parabola occurs at $x=-\frac{b}{2a}$. Substituting the values of $a$ and $b$ into the formula, we have $x=-\frac{18}{2\times(-2)}=\frac{18}{4}=4.5$ feet.

Step1: First, expand $f(x)$.

$f(x)=(x + 5)(x - 3)=x^{2}+2x-15$. Then $g(x)=x\cdot f(x)=x(x^{2}+2x - 15)=x^{3}+2x^{2}-15x$.

Step2: Find the x - intercepts.

The x - intercepts of $g(x)$ are found by setting $g(x)=0$. So $x^{3}+2x^{2}-15x = 0$. Factor out an $x$: $x(x^{2}+2x - 15)=0$. Then factor the quadratic: $x(x + 5)(x - 3)=0$. The solutions of the equation are $x = 0$, $x=-5$, and $x = 3$. So the function $g(x)$ has 3 x - intercepts.

Step1: Calculate the sum, difference, product and quotient of $f(x)$ and $g(x)$.

$f(x)=x^{2}$ and $g(x)=3x - 2$.

• $f(x)+g(x)=x^{2}+3x - 2$.
• $f(x)-g(x)=x^{2}-(3x - 2)=x^{2}-3x + 2$.
• $f(x)\cdot g(x)=x^{2}(3x - 2)=3x^{3}-2x^{2}$.
• $f(x)\div g(x)=\frac{x^{2}}{3x - 2}$ (for $x

eq\frac{2}{3}$).
By observing the graph of $h(x)$ which passes through the origin and has a non - quadratic shape, we can see that $h(x)=f(x)\cdot g(x)=x^{2}(3x - 2)=3x^{3}-2x^{2}$.

Answer:

b. 4.5 feet

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