Question

the rate of photosynthesis r for a certain plant depends on the intensity of light x, in lumens, according to ( r(x)=480x - 80x^{2} ).
a. sketch the graph of this function on a meaningful window.
b. determine the intensity x that gives the maximum rate of photosynthesis.
a. choose the correct graph below.

Explanation:

Step1: Analyze the function type

The function \( R(x)=480x - 80x^{2}\) is a quadratic function. The general form of a quadratic function is \(y = ax^{2}+bx + c\), where \(a=-80\), \(b = 480\), \(c = 0\). Since \(a=-80<0\), the parabola opens down - ward.

Step2: Find the x - intercepts

Set \(R(x)=0\), then \(480x-80x^{2}=0\). Factor out \(80x\): \(80x(6 - x)=0\).
Using the zero - product property \(80x=0\) gives \(x = 0\), and \(6 - x=0\) gives \(x = 6\).

Step3: Find the vertex

The x - coordinate of the vertex of a quadratic function \(y = ax^{2}+bx + c\) is given by \(x=-\frac{b}{2a}\).
Substitute \(a=-80\) and \(b = 480\) into the formula: \(x=-\frac{480}{2\times(-80)}=\frac{-480}{-160}=3\).
When \(x = 3\), \(R(3)=480\times3-80\times3^{2}=1440 - 720=720\).

Step4: Analyze the options

• Option A: The parabola opens upward (\(a>0\)), which is incorrect since \(a=-80<0\).
• Option B: The x - intercepts are not \(x = 0\) and \(x = 6\) (the right - hand x - intercept is \(x = 8\) in this option), so it is incorrect.
• Option C: The parabola opens downward (\(a<0\)), has x - intercepts at \(x = 0\) and \(x = 6\), and the vertex at \(x = 3\) (since the function is symmetric about \(x = 3\) for \(y=-80x^{2}+480x\)).
• Option D: The parabola opens upward (\(a>0\)), which is incorrect.

Step5: Solve for the maximum rate (part b)

For a quadratic function \(y = ax^{2}+bx + c\) (\(a
eq0\)), the x - value of the vertex gives the maximum (when \(a<0\)) or minimum (when \(a>0\)) value.
Using the formula \(x=-\frac{b}{2a}\) with \(a=-80\) and \(b = 480\), we have \(x = 3\).

Answer:

a. C.
b. \(x = 3\) lumens.