Question 750·Medium·Math

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The luminosity, in watts, of a certain LED bulb $t$ hours after it is switched on, where $t\ge0$ , can be modeled by the function

L(t)=-0.04t^2+0.8t+6

Based on this model, after how many hours will the bulb reach its greatest luminosity?

A quadratic model with a negative squared coefficient graphs as a downward-opening parabola. Its highest point is the vertex, the graph's turning point. Graph the function in Desmos and click that point. Read the horizontal coordinate for the time of the maximum, not the vertical coordinate for the maximum output. Check that the time is allowed.

Hints

Hint 1
The coefficient on $t^2$ is negative, so the graph bends downward. A downward-opening parabola has its maximum at its vertex, the highest turning point. Which point on the graph should you inspect?
Hint 2
A point on this graph is written $(t,L(t))$ . The first coordinate counts hours since switching on; the second gives luminosity in watts. Once you find the peak, which coordinate answers the question?

Step-by-step

Graph the model and read its peak
Step 1
Identify where the greatest luminosity occurs
The coefficient of $t^2$ is $-0.04<0$ , so the graph is a downward-opening parabola, the curve made by a quadratic function. Its highest point, called the vertex, gives the greatest luminosity.
Step 2
Read the time at the peak
Type $L(x)=-0.04x^2+0.8x+6$ in Desmos, using $x$ for the time $t$ . Click the peak; Desmos labels it $(10,10)$ . The first coordinate is the time, while the second is the luminosity in watts. Since $10\ge0$ , the time is allowed. The bulb reaches its greatest luminosity $10$ hours after it is switched on. Choice C.

Question 750·Medium·Math

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