Question

as part of a recreational waterfront grant, the city council plans to enclose a rectangular area along the waterfront of a lake and create a park and swimming area. the budget calls for the purchase of 3,000 feet of fencing. (note: there is no fencing along the lake.) an equation that expresses the area a of this rectangular site as a quadratic function of x is a(x) = -2x² + 3,000x. complete parts a through c below. a. use your graphing calculator to graph the area function. each graph below has the viewing window 0,1500,150×0,2000000,200000. choose the correct graph below. ○ a. ○ b. ○ c. ○ d.

Explanation:

The function \( A(x) = -2x^{2}+3000x \) is a quadratic function with a negative leading coefficient (\( a = - 2<0 \)), so its graph is a parabola opening downward. We can also find the vertex of the parabola. The x - coordinate of the vertex of a quadratic function \( y=ax^{2}+bx + c \) is given by \( x=-\frac{b}{2a} \). For \( A(x)=-2x^{2}+3000x \), \( a=-2 \) and \( b = 3000 \), so \( x=-\frac{3000}{2\times(-2)}=\frac{3000}{4} = 750 \). The y - coordinate (area) at \( x = 750 \) is \( A(750)=-2\times(750)^{2}+3000\times750=-2\times562500 + 2250000=- 1125000+2250000 = 1125000 \). The graph should start from the origin (when \( x = 0 \), \( A(0)=0 \)) and rise to a maximum at \( x = 750 \) and then fall back to 0 (when \( A(x)=0 \), \( -2x^{2}+3000x=0\Rightarrow x(-2x + 3000)=0\Rightarrow x = 0 \) or \( x = 1500 \)). Among the given graphs, option B shows a parabola opening downward, starting from the origin, rising to a peak, and then falling back to 0 at \( x = 1500 \), which matches the properties of the function \( A(x) \).

Answer:

B.