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Chapter 4: Problem 9

Find the dimensions of the rectangle with maximum area that can be inscribed in a circle of radius 10 .

Short Answer

Expert verified
The rectangle with maximum area is a square with dimensions \(10\sqrt{2} \times 10\sqrt{2}\).

Step by step solution

01

Understanding the Problem

We want to find the rectangle with the largest possible area that can be inscribed inside a circle with a radius of 10. This means that all four corners of the rectangle must lie on the circle.
02

Identify the Relationship Between Radius and Dimensions

If we inscribe a rectangle within a circle, the diameter of the circle becomes the diagonal of the rectangle. Since the radius is given as 10, the diameter is 20, so we have: \ a^2 + b^2 = 20^2, where \(a\) and \(b\) are the width and height of the rectangle.
03

Express Area in Terms of One Variable

We want to maximize the area \(A\) of the rectangle, given by \(A = a \times b\). From the equation \(a^2 + b^2 = 400\), solve for \(b\): \(b^2 = 400 - a^2\) and thus \(b = \sqrt{400 - a^2}\). Now, \(A = a \times \sqrt{400 - a^2}\).
04

Maximize the Area Function

To find the maximum area, differentiate \(A\) with respect to \(a\) and set the derivative equal to zero to find critical points. \ \(\frac{dA}{da} = \sqrt{400 - a^2} - \frac{a^2}{\sqrt{400 - a^2}} = 0\). Simplify the equation to solve for \(a\).
05

Find the Critical Point

Through solving the derivative equation, we find that when \(a = \sqrt{200}\), \(b = \sqrt{200}\), making the dimensions of the rectangle \(\sqrt{200} \times \sqrt{200}\), which is a square.
06

Conclusion - Dimensions of the Rectangle

The rectangle with maximum area that can be inscribed in the circle is actually a square with the dimensions \(\sqrt{200} \times \sqrt{200}\), simplifying to \(10\sqrt{2}\) by calculating \(\sqrt{200} = 10\sqrt{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Maximum Area
When tackling any optimization problem in mathematics, understanding how to maximize or minimize a function is key. In our context, we're asked to maximize the area of a rectangle inscribed within a circle. The largest area means that we're looking for a specific arrangement of rectangle dimensions to yield the greatest product of its sides.
  • The rectangle's four corners touching the circle tells us the maximum span across the circle.
  • The formula for area, \(A = a \times b\), needs to be reworked using relationships derived from the circle's properties.
By expressing one variable in terms of another, we simplify the expression for maximum area. The challenge lies in using calculus to identify where this maximum value occurs. Through differentiation, we identify critical points that help us pinpoint the maximum area configuration.
Inscribed Rectangle
An inscribed rectangle is a figure drawn within another shape such that all its corners rest on the boundary of the larger shape. In this case, our rectangle is inscribed within a circle.
  • The circle acts as a boundary limit, clearly restricting the dimensions of the rectangle.
  • For a rectangle inscribed in a circle, the circle's diameter will be equal to the diagonal of the rectangle.
For maximizing the area of the rectangle, we see that it ultimately forms a square. This happens because a rectangle with equal side lengths (a square) will naturally have its diagonal as the maximum possible, aligning perfectly with the circle's diameter. This is why, often in inscribed rectangle problems, the solution leads to a square formation within the circle.
Circle Geometry
Understanding the geometric properties of a circle is crucial for problems like this one. The most salient point here is how the circle's radius and diameter define the possible dimensions of an inscribed rectangle.
  • The circle鈥檚 diameter is twice the radius, providing us with a central key: the length of the diagonal of any rectangle inscribed in the circle.
  • The Pythagorean theorem is instrumental, where for any rectangle side lengths \(a\) and \(b\), we have \(a^2 + b^2 = (2r)^2\) with \(r\) as the radius.
This geometry reveals the natural mathematical relationships between shapes. By thoroughly understanding this, we are able to reformulate expressions to find maximum areas, optimize designs, and appreciate the inherent symmetry within geometric configurations.

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Most popular questions from this chapter

A wire of length 12 in can be bent into a circle, bent into a square, or cut into two pieces to make both a circle and a square. How much wire should be used for the circle if the total area enclosed by the figure(s) is to be (a) a maximum (b) a minimum?

A firm determines that \(x\) units of its product can be sold daily at \(p\) dollars per unit, where $$ x=1000-p $$ The cost of producing \(x\) units per day is $$ C(x)=3000+20 x $$ (a) Find the revenue function \(R(x)\). (b) Find the profit function \(P(x)\). (c) Assuming that the production capacity is at most 500 units per day, determine how many units the company must produce and sell each day to maximize the profit. (d) Find the maximum profit. (c) What price per unit must be charged to obtain the maximum profit?

Prove: If \(f(x) \geq 0\) on an interval and if \(f(x)\) has a maximum value on that interval at \(x_{0}\), then \(\sqrt{f(x)}\) also has 芒 maximum value at \(x_{0} .\) Similarly for minimum values. [Hint: Use the fact that \(\sqrt{x}\) is an increasing function on the interval \([0,+\infty) .\) ]

A position function of a particle moving along a coordinate line is provided. (a) Evaluate \(s\) and \(v\) when \(a=0 .\) (b) Evaluate \(s\) and \(a\) when \(v=0\). $$ s=\ln \left(3 t^{2}-12 t+13\right) $$

Suppose that the position functions of two particles, \(P_{1}\) and \(P_{2}\), in motion along the same line are $$ s_{1}=\frac{1}{2} t^{2}-t+3 \text { and } s_{2}=-\frac{1}{4} t^{2}+t+1 $$ respectively, for \(t \geq 0\). (a) Prove that \(P_{1}\) and \(P_{2}\) do not collide. (b) How close do \(P_{1}\) and \(P_{2}\) get to each other? (c) During what intervals of time are they moving in opposite directions?
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