Area Under a Line
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The figure shows a rectangle of dimensions
and
drawn under the line
a) Find the location
of the point associated with the largest area, and b) Find the area of the largest rectangle.
What do we know about
and
?
The point is on the given line, so these coordinates must satisfy the equation of the line.
Express this fact.
Equation 1
In order to find the maximum area, A, we need to express the area in terms of
and
?
Do that.
Equation 2
How can we find the maximum area?
We need to express the area in terms of one variable and differentiate.
Combine equations 1 and 2 to express the area in terms of
Equation 3
Differentiate.
Equation 4
Find the value of x1 which produces a maximum.
How do we find the corresponding value of y1?
We can substitute into equation 1.
Do that.
We get
We now have determined the coordinates (3, 1) required in part (a) of the problem.
How do we find the maximum area for part (b)?
We use these coordinates in equation 2. Do that.
Here is a graph showing this maximum area.
How do we check if this is a maximum and not a minimum?
We can take the second derivative of the area and check its sign when evaluated at x = 3. Which sign will indicate a maximum?
A negative sign is associated with a maximum, where the slope is changing toward less positive values.
Take the second derivative, using equation 4.
Do we have a maximum?
Yes, because this is negative, independent of the value of x.
One more way to check is to draw a graph of area versus x using equation 3. Do that on your own and check with the figure below.
Graph of Area vs the x-coordinate of the corner point.
A local maximum shows at x = 3.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor