The rate of photosynthesis R for a certain plant depends on the intensity of light x, in lumens, according to Upper R left parenthesis x right parenthesis equals 450 x minus 90 x squared.

Question

The rate of photosynthesis R for a certain plant depends on the intensity of light​ x, in​ lumens, according to Upper R left parenthesis x right parenthesis equals 450 x minus 90 x squared.
 The intensity x that gives the maximum rate of photosynthesis is 
  
enter your response here ​lumen(s).

GPT-4o mini · Answer

To find the intensity $ x $ that gives the maximum rate of photosynthesis, we need to find the maximum of the function $ R(x) = 450x - 90x^2 $. This is a quadratic function in the form of $ R(x) = ax^2 + bx + c $, where $ a = -90 $ and $ b = 450 $. The maximum of a quadratic function occurs at the vertex, which can be found using the formula: $$ x = -\frac{b}{2a} $$ Substituting in our values for $ b $ and $ a $: $$ x = -\frac{450}{2 \cdot -90} = -\frac{450}{-180} = \frac{450}{180} = 2.5 $$ Thus, the intensity $ x $ that gives the maximum rate of photosynthesis is: $$ \boxed{2.5} \text{ lumen(s)} $$

The rate of photosynthesis R for a certain plant depends on the intensity of light​ x, in​ lumens, according to Upper R left parenthesis x right parenthesis equals 450 x minus 90 x squared.

The rate of photosynthesis R for a certain plant depends on the intensity of light x, in lumens, according to Upper R left parenthesis x right parenthesis equals 450 x minus 90 x squared.