### Problem StatementA rectangle has its right side on the line $ x = 6 $ and its base on the x-axis. Its top left corner lies on the graph of $ y = \sqrt{x} $.**Tasks:**#### a) Draw a picture of the described scenario and label any important lengths.**Explanation of Diagram:**There is a coordinate plane with an $ X $-axis and $ Y $-axis. On this plane, the graph of the function $ y = \sqrt{x} $ is drawn, which is a curve starting from the origin (0,0) and moving up to the right. A rectangle is depicted with its right side on the vertical line $ x = 6 $ and its base along the x-axis, indicating that its height comes from the point on the function $ y = \sqrt{x} $.#### Graph Description:- The x-axis and y-axis, both marked with arrows pointing in the positive direction.- A curve corresponding to the function $ y = \sqrt{x} $, starting at the origin (0,0) and curving upwards.- A rectangle with one of its sides touching the line $ x = 6 $ and its base along the x-axis.- The top left corner of the rectangle lies on the curve $ y = \sqrt{x} $.#### b) Find the dimensions of the rectangle that maximizes its area. The goal here is to determine the dimensions of this rectangle such that its area is maximized, given that its right side is fixed at $ x = 6 $ and its top left corner lies on the curve $ y = \sqrt{x} $.**Note:**To solve this problem, you need to express the area of the rectangle as a function of one variable, use calculus to find the maximum value of this function, and derive the dimensions from this maximum value.

Answered: Image /qna-images/answer/205a0bbf-a8c2-4a70-8a9d-e751412562f4.jpg

a rectangle has its rignt side on the line X = Le and its base on the x-axiis. Its top left comer lies on the of 4=VX. graph + label oany umportant a) draw a pichure of the described scenano lengths. b) find the that maximizes its area., dimensions of the rectangle

a rectangle has its rignt side on the line X = Le and its base on the x-axiis. Its top left comer lies on the of 4=VX. graph + label oany umportant a) draw a pichure of the described scenano lengths. b) find the that maximizes its area., dimensions of the rectangle

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

### Problem Statement

A rectangle has its right side on the line $ x = 6 $ and its base on the x-axis. Its top left corner lies on the graph of $ y = \sqrt{x} $.

**Tasks:**

#### a) Draw a picture of the described scenario and label any important lengths.

**Explanation of Diagram:**

There is a coordinate plane with an $ X $-axis and $ Y $-axis. On this plane, the graph of the function $ y = \sqrt{x} $ is drawn, which is a curve starting from the origin (0,0) and moving up to the right. A rectangle is depicted with its right side on the vertical line $ x = 6 $ and its base along the x-axis, indicating that its height comes from the point on the function $ y = \sqrt{x} $.

#### Graph Description:

- The x-axis and y-axis, both marked with arrows pointing in the positive direction.
- A curve corresponding to the function $ y = \sqrt{x} $, starting at the origin (0,0) and curving upwards.
- A rectangle with one of its sides touching the line $ x = 6 $ and its base along the x-axis.
- The top left corner of the rectangle lies on the curve $ y = \sqrt{x} $.

#### b) Find the dimensions of the rectangle that maximizes its area.

The goal here is to determine the dimensions of this rectangle such that its area is maximized, given that its right side is fixed at $ x = 6 $ and its top left corner lies on the curve $ y = \sqrt{x} $.

**Note:**

To solve this problem, you need to express the area of the rectangle as a function of one variable, use calculus to find the maximum value of this function, and derive the dimensions from this maximum value.

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