1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
wel
2 years ago
10

Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r. width units height units

Mathematics
1 answer:
dmitriy555 [2]2 years ago
6 0

This question is incomplete, the remaining part of the question is upload as an image alongside this answer.

Answer:

Width = 2x = 2( r/√2 ) =  √2 r units

Height = 2y = 2( r/√2 ) =  √2 r units

Step-by-step explanation:

From the Figure on the image; lets consider the circle of radius r, centered at the origin.  

let ABCD be the largest rectangle that can be inscribed inside the circle.  

Let the half width of the rectangle be x, then in right triangle ONB using Pythagorean theorem,  

half height of rectangle y = √(r² - x²)

Thus the width of the inscribed rectangle = 2x and height of the inscribed rectangle = 2√(r² - x²)

thus the area of the inscribed rectangle = length × width

⇒ A(x) = 2x(2√(r² - x²))

⇒ A(x) = 4x√(r² - x²)  

now in order to maximize the area, we find critical points.

so we find the derivative and set that zero, that is Ai(x) = 0.

so using product rule, we get

A'(x) = 4x × ( -2x / 2√(r² - x²) ) + ( 4 × √(r² - x²)  )

A'(x) = ( -4x² / √(r² - x²) ) + ( 4√(r² - x²)  )

Now for critical points, set A'(x) = 0

so

( -4x² / √(r² - x²) ) + ( 4√(r² - x²)  ) = 0

( 4x² / √(r² - x²) ) =  ( 4√(r² - x²)  )

x² = r² - x²

2x² = r²

x² = r²/2

x = ±√(r²/2)

Now since x represent the with, it cannot be negative, Thus

x = r/√2

hence

y = √(r² - x²) = √(r² -  r²/2) = √(r²/2) =  r/√2

Therefore, the dimensions of the rectangle of largest area will be;

Width = 2x = 2( r/√2 ) =  √2 r units

Height = 2y = 2( r/√2 ) =  √2 r units

You might be interested in
Two-thirds a number plus 4 is 7
ICE Princess25 [194]

The equation (form) is: \frac{2}{3}x + 4 = 7

The solution of the equation is: x = \frac{9}{2}


Explanation:

First, let's call the number we are going to solve for "x".

Then two thirds of this number can be written as:

\frac{2}{3}x

Then "plus 4" means add 4 we get:

\frac{2}{3}x + 4

"is 7" means that the above expression is equal to 7 (as follows):

\frac{2}{3}x + 4 = 7

The solution of the above equation is:

\frac{2}{3}x + 4 = 7 \\ \frac{2}{3}x = 3 \\ x =  \frac{9}{2}

6 0
2 years ago
Which is grater to least 56.01 56.10 or 56.011
lapo4ka [179]
The answer would be 56.10
4 0
3 years ago
Read 2 more answers
Rewrite 39,500 × 103 in scientific notation.<br><br> Thank you
Mice21 [21]

Step-by-step explanation:

39500 x 103 = 4,068,500

in scientific notation, that ia 4.0685 x 10⁶

8 0
2 years ago
How would you write a decimal that is less than five Ten thousandths
asambeis [7]

Five ten-thousandths looks like this:  0.0005

All of these are less:

0.0004
0.0003
0.0002
0.0001
0.00001
0.000001
.
.
etc.

3 0
3 years ago
PLEASEEE I NEED HELP ASAPPPPPP DUE IN 6 MINS! PLEASE
sergey [27]

Step-by-step explanation:

60x + 3300 = 24000

60x = 24000 - 3300

60x = 20700

x = 20700 ÷ 60

x = 345

x ≤345

5 0
2 years ago
Other questions:
  • Find the zeros of polynomial functions and solve polynomials equations.<br> f(x)=2x^3+2x^2-4x
    5·1 answer
  • PLS HELP ME ASAP
    8·1 answer
  • A road construction crew is resurfacing the highway between point d and e. The distance between point d and e is 93 miles of roa
    13·1 answer
  • What did I do wrong when finding out the sector area? I know the answer I got is wrong so can someone please help me
    6·1 answer
  • Complete the equation of the line whose slope is 5 and y-intercept is (0,4)
    12·2 answers
  • given the rule s=t+3 and the starting number 0, create a table to show the first 5 terms in the sequence graph the resulting ord
    12·1 answer
  • What are the roots of x^3-8=0?
    8·1 answer
  • Stacy exercises 3 days each week by walking around the soccer field near her home the field is 80 yards wide and 115 yards long
    12·1 answer
  • 423 times 28 in grid method
    5·1 answer
  • PLEASE HELP I NEED THIS QUICK IM SO STUCK.<br>​
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!