A rectangle is constructed with its base on the x-axis and two of its vertices o

regatamin2

regatamin2

Answered question

2021-12-06

A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y=16x2
What are the dimensions of the rectangle with the maximum area? What is that area?

Answer & Explanation

Laura Worden

Laura Worden

Beginner2021-12-07Added 45 answers

Step 1

Step 2
The problem
A=2xymax
given
y=16x2
So, we need
2x(16x2)=f(x)max
We havef(x)=(32x2x3)=326x2
Equating f(x)=0x=163=43 is the positive solution.
Since f(x) changes its sign from positive to negative when passing through x=43,there is a maximum at that point.
Corresponding y is
y=16x2=16163=323
so the dimensions are
83×323
and the area is
A=83×323=25633
Dimensions: 83×323, Area 323
Janet Young

Janet Young

Beginner2021-12-08Added 32 answers

Step 1
A Rectangle with its base on x-axis and two of its vertices on the parabola whose equation is y=16x2
Let A be the area of a rectangle, the expression is given as:
A=2xymax
Now, we need to maximize the area of a rectangle that is,f(x)=2x(16x2)
To maximize the area, differentiate the above function with respect to x and equate it with zero,
f(x)=326x2
326x2=0
x=163
=43
2.31
Put the value of x in the equation of parabola and find the value of y,
y=16x2
y=16163
y=323
10.67
In the rectangle with the maximum area, the shorter dimension is 2.31 and longer dimension is 10.76
Thus, the maximum area of the rectangle is,
A=2×2.31×10.76
49.29sq. units

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?