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Calculus Word + Linear Problem (2)

  • Thread starter sheet1
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  • #1
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hello I am new here and just starting to do some maths in school (not always my strong point) and I am need of help for some problems that I am trying to solve!

Homework Statement



A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 37 feet, express the area of the window as a function of the width (across the base) of the window.

a(x) = ?

Homework Equations



some equations used are x*y = area or 2x + 2y and πr^2

The Attempt at a Solution



from my calculations I thought the answers would be 37x-x^2, but apparently it isn;t, and I dont have any solutions to confirm my answer(s). THis probably seems easy to you guys, but word problems have me clueless.

Homework Statement



THis second question is a linear matrix algebra type of question...

Solve the system:

1 1 0 0 : -1
0 1 1 0 : 3
0 0 1 1 : -1
1 0 0 1 : -5

Now normally i know how to solve this but this time its asking for the solution in the form of this and dont know how to input the values:

[x1] = [] []
[x2] = [] + [] s.
[x3] = [] []
[x4] = [] []


Homework Equations



Regular algebra...but haven't seen this type of question yet so there is perhaps something i dont know?

The Attempt at a Solution



I attempted solving it such as a normal matrix and then solving for x1,x2 etc...and after inputting the numbers my answers was incorrect.

i had x1 = -1 + -1 x2 = 3 + -1 x3 = -1 + -1 x4 = 1 -4
 
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Answers and Replies

  • #2
tiny-tim
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hello sheet1! welcome to pf! :smile:
A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 37 feet, express the area of the window as a function of the width (across the base) of the window.
what is your formula for L (the length of the rectangle) as a function of W (the width?)
Solve the system:

1 1 0 0 : -1
0 1 1 0 : 3
0 0 1 1 : -1
1 0 0 1 : -5
i don't understand what this means :confused:

(the solution looks like a constant vector V that would be a solution to a "zero" equation, plus a parameter s times a particular solution vector W: V + sW)
 
  • #3
SteamKing
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First, you've got to figure how the perimeter of the window relates to ts dimensions.
 
  • #4
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Ok I edited the first one and added a(x)= ? ... thats all the information I have on that question, and for the second one its hard to describe since I can't add a picture:

its formatted as such:

x1 + x2 = -1
x2 + x3 = 3
x3 + x4 = -1
x1 + x4 = -5

then there is 3 groups in a square bracket, one with x1 2,3,4 in a vertical line equaling another square box with 4 vertical blank solutions following by that plus (+) another 4 vertical boxes, the question is very similar to this youtube video, i followed it but could not get a correct answer...:

 
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  • #5
Ray Vickson
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Ok I edited the first one and added a(x)= ? ... thats all the information I have on that question, and for the second one its hard to describe since I can't add a picture:

its formatted as such:

x1 + x2 = -1
x2 + x3 = 3
x3 + x4 = -1
x1 + x4 = -5

then there is 3 groups in a square bracket, one with x1 2,3,4 in a vertical line equaling another square box with 4 vertical blank solutions following by that plus (+) another 4 vertical boxes, the question is very similar to this youtube video, i followed it but could not get a correct answer...:

So, what steps have you taken in solving the system? You are required to actually _show_ what you did.
 
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  • #6
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for the first question I did something like this (I have lots of notes scribbles, many attempts, easier to say what I did rather than write stuff out here, since it is much harder):

perim=x+2*h+Pi*x=37 => h=(37-x-Pi*x)/2
A(x)=h*x+Pi*x^2=(37-x-Pi*x)*x+Pi*x^2=37x-x*x

for the systems I just isolated the values and solved like normally, but what I DONT understand is what they are asking for in the question, the video i showed above shows it in detail of how to solve it, but after following it step by step I keep getting a wrong answer and am unaware on how to solve a system described in such a way...i need help! i just want to learn..
 
  • #7
tiny-tim
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perim=x+2*h+Pi*x
ah, nooo :redface:

perim = x + 2*h + Pi*x/2 :wink:
 
  • #8
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ok thanks for that, i got the solution! -.8926990813x^2 + 18.5x ... now i really need help with the linear problem, please with a cherry on top with chocolate whipped cream.
 
  • #9
tiny-tim
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for the systems I just isolated the values and solved like normally, but what I DONT understand is what they are asking for in the question, the video i showed above shows it in detail of how to solve it, but after following it step by step I keep getting a wrong answer and am unaware on how to solve a system described in such a way...i need help! i just want to learn..
ok, show us how far you got :smile:
 
  • #10
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0
ok i got this far :

a = -1 -b
b = 3 -c
c = -1 -d
d = -5 -a

a being x1 b being x2 and so forth...just not sure what to do from this point on. If I plug one of the values into another they just cancel out each other, was I supposed to use gauss-Jordan elimination even further?
 
  • #11
237
5
ok thanks for that, i got the solution! -.8926990813x^2 + 18.5x
I can tell you've worked out how to get the answer but your 10th significant figure is wrong!
 
  • #12
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not according to the computer it aint! oay, do u think you could shed some light on my linear equation?
 
  • #13
SteamKing
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I don't think the question asks you to find the numerical value of the width, just express the area A of the window in terms of the width W, subject to the constraint that the perimeter of the window is 37 feet.

Since the window is a rectangle with a semi-circle on top, one can see that the diameter of the semi-circle D must be equal to the width W of the rectangle below.

The perimeter of the semi-circle is
π * D / 2, or π * W / 2
The perimeter of the rectangle below is
W + 2 * H

The total perimeter of the window is P =
π * W / 2 + W + 2 * H
= 37 feet

P =
(1 + π/2)*W + 2*H
= 37 feet

Solve for H.

Then, calculate the area of the semi-circle and the rectangle and add them together.

I have no idea why you wrote all those linear equations to solve.
 
  • #14
Ray Vickson
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I don't think the question asks you to find the numerical value of the width, just express the area A of the window in terms of the width W, subject to the constraint that the perimeter of the window is 37 feet.

Since the window is a rectangle with a semi-circle on top, one can see that the diameter of the semi-circle D must be equal to the width W of the rectangle below.

The perimeter of the semi-circle is
π * D / 2, or π * W / 2
The perimeter of the rectangle below is
W + 2 * H

The total perimeter of the window is P =
π * W / 2 + W + 2 * H
= 37 feet

P =
(1 + π/2)*W + 2*H
= 37 feet

Solve for H.

Then, calculate the area of the semi-circle and the rectangle and add them together.

I have no idea why you wrote all those linear equations to solve.
He said it was a second question. He just mis-formatted his posting.
 

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