Help with Functions: Find the Area of a Norman Window

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A Norman window consists of a rectangle topped by a semicircle, and the perimeter is given as 30 feet. The initial equation for the perimeter, 2x + 2y = 30, is incorrect as it does not account for the semicircular part. The correct approach involves adjusting the perimeter equation to include the semicircle, which requires dividing the semicircle's area and perimeter by two. The area function should be expressed as A = x(15/2 - x) + (1/2)π(x/2)^2. Clarification on these equations is essential to avoid errors in calculations.
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Help with Functions!

Hi everone this is my first thread and I am looking forward to particitate in PF. Anyways I need help with a question...although I found a solution but I am not sure if it is right..any help would be greatly appreaciated.
Question - A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30ft, express the area A of the window as a function of the width x of the window.
Solution - 2x+2y=30, solvin for y; y=15/2-x
A=Rectangle + semicircle
A=xy + "pi"(x/2)^2
A=x(15/2-x)+"pi"x^2/4
A=15x/2-x^2+"pi"x^2/4
Well...it looks pretty simple but last assignment I did, I got a low mark and I screwed up in many places...so I just want to make sure it doesn't happen again...thanks in advance.
 
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Hi the game, warm welcome to PF ! You have found homework help paradise. :biggrin:

I'm afraid I disagree with your first equation :eek:.

You give 2x+2y=30 as the equation of the perimeter. But this looks to me like the equation for the perimeter of a rectangular window! Try working out a new perimeter equation considering the circular higher part of the window.

And keep in mind when you use the equation of perimeter and area of a circle that you are dealing with a semicircle. Hence you must divide these expressions by half! Ex: the equation of the area and perimeter of the semi-circle are respectively

A = \frac{1}{2}\pi r^2

P = \pi r
 
Thanks quasar987...I really appreaciate your help...I love PF homework help :)
 
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Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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