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
notsponge [240]
3 years ago
7

A Norman window has the shape of a rectangle surmounted by a semicircle. Suppose the outer perimeter of such a window must

Mathematics
1 answer:
Feliz [49]3 years ago
6 0

The base length that will maximize the area for such a window is 168.03 cm. The exact largest value of x when this occurs is 233.39 cm

Suppose we make an assumption that:

  • (x) should be the width of the rectangle base;
  • (h) should be the height of the rectangle

Also, provided that the diameter of the semi-circle appears to be the base of the rectangle, then;

  • the radius  \mathbf{r = \dfrac{x}{2}}  

and, the perimeter of the window can now be expressed as:

\mathbf{x + 2h + \pi r = x + 2h + \dfrac{\pi x }{2}}

\mathbf{= \Big ( 1 + \dfrac{\pi}{2}\Big) x + 2h}

Given that the perimeter = 600 cm

∴

\mathbf{ \Big ( 1 + \dfrac{\pi}{2}\Big) x + 2h= 600}

\mathbf{  h = 300 - \Big( \dfrac{1}{2} + \dfrac{\pi}{4}\Big) x}

Since h > 0, then:

\mathbf{  h = 300 - \Big( \dfrac{1}{2} + \dfrac{\pi}{4}\Big) x>0}

By rearrangement and using the inverse rule:

\mathbf{  x<  \dfrac{ 300}{\Big( \dfrac{1}{2} + \dfrac{\pi}{4}\Big) } }

\mathbf{  x=  \dfrac{ 1200}{\Big( 2 +\pi \Big) } }

\mathbf{  x=  233.39 \ cm }

Thus, the largest length x = 233.39 cm

However, the area of the window is given as:

\mathbf{A(x) = xh + \dfrac{1}{2} \pi r^2}

\mathbf{A = x \Big [  300 - \Big ( \dfrac{1}{2}+\dfrac{1}{4} \Big) x \Big ]  +\dfrac{1}{2}\pi \Big(\dfrac{x}{2} \Big )^2}

\mathbf{A (x) = 300x - \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big) x^2 \ cm^2}

Now, at maximum, when the area A = 0. Taking the differentiation, we have:

\mathbf{\dfrac{d}{dx} 300x - \dfrac{d}{dx} \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big) x^2 \ =0}

\mathbf{ 300 - 2x \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big)  \ =0}

Making x the subject of the formula, we have:

\mathbf{x = \dfrac{1200}{4 +\pi}}

x = 168.03 cm

Taking the second derivative:

\mathbf{\dfrac{d}{dx} \Big [300 -2x \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big) \Big]}

\mathbf{= -2 \Big( \dfrac{1}{2}+\dfrac{\pi}{8}\Big )

Therefore, we can conclude that the maximum area that exists for such a window is 168.03 cm

Learn more about derivative here:

brainly.com/question/9964510?referrer=searchResults

You might be interested in
A sequence is defined recursively by the formula f(n + 1) = f(n) + 3 . The first term of the sequence is –4. What is the next te
sergiy2304 [10]
Translate the equation to math.
It says the term after the current term is the current term plus 3.
Next term = this term + 3
Next term = -4+3
The next term is then -4+3 or -1.

Answer= - 1

Hope this helps :)
7 0
3 years ago
???????????????????????????????????????
kolezko [41]
3',8 30,80 7,2 (13,18
5 0
3 years ago
On Tuesday, Michael walked to a grocery store in the morning and decided to buy a banana for $9.12 Michael handed the salesperso
mixas84 [53]
Michael received $0.01 or a penny
8 0
3 years ago
There are two colored marbles and one white marble in a box. Julie mixes the marbles and then Sam draws two marbles at random wi
tino4ka555 [31]
At first there are three marbles for Sam to pick from. the possibility of drawing a colored marble is 2/3. if Sam picks one of the colored marbles, the possibility of therefore (if he'd drawn the colored marble) drawing the second colored marble is 1/2. for the possibility of those happening consequently, we have to multiply the possibilities. 2/3 * 1/2 is 2/6 is 1/3. which means Sam wins in 33.333...% cases. Julie has a better chance of winning.

adding a white marble would change the possibilities to 1/2 and 1/3 consequently, meaning Sam wins in 1/2 * 1/3 = 1/6 = 16.7% cases, which is even less fair.

adding a colored marble would change the possibilities to 3/4 and 2/3 consequently, meaning Sam wins in 3/4 * 2/3 = 6/12 = 50% cases, which finally makes the game fair.
6 0
1 year ago
What of following is the inverse of y=6
Elenna [48]

Answer:

x=6

Step-by-step explanation:

The inverse is the equation with the x and y variables transposed

5 0
3 years ago
Other questions:
  • Value of 6n-2 when n=3
    6·2 answers
  • 25 points if you answer please answer
    8·2 answers
  • To avoid the pond, Fred must walk 10 meters south and 24 meters east. To the nearest meter, how many meters would be saved if it
    9·1 answer
  • WIll reward with high brainly points if these questions are aswered correctly
    5·1 answer
  • Using the linear combination method, what is the solution to the system of linear equations and ? (3, 2) (2, 3) (2, 3) (3, 2)
    15·2 answers
  • Is the function y = 6x - 10<br><br> linear<br><br> Or <br><br> nonlinear
    10·2 answers
  • Pls help LOTS OF POINTS!!!! HELP!!!!!!
    8·2 answers
  • Whats 1 plus 1 please help​
    6·1 answer
  • How long is a bus journey that starts at 8.55 a.m. and ends at 9.17 a.m.?
    13·1 answer
  • A cell phone towerbcast a shadow that is 40 feet long . An 10- foot- tall stop sign located near the tower casts a shadow that i
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!