Expressing functions [word problems]

**Aravsion** · September 17th 2011, 08:37 PM

Could somebody help me understand the process of solving these problems... Also, could someone provide a link to a site where I can practice more of these type of problems.
1) The problem statement, all variables and given/known data:
A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108". Express the volume of such a package as a function of x.

2) The problem statement, all variables and given/known data:
A Palladian window is a rectangle with a semicircle on top, as shown. Suppose the perimeter of a particular Palladian window is 28 ft. Express the total area of the window as a function of x.

Expressing functions [word problems]-palladian-window.jpg

**Prove It** · September 17th 2011, 10:46 PM

Originally Posted by Aravsion

Could somebody help me understand the process of solving these problems... Also, could someone provide a link to a site where I can practice more of these type of problems.
1) The problem statement, all variables and given/known data:
A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108". Express the volume of such a package as a function of x.

2) The problem statement, all variables and given/known data:
A Palladian window is a rectangle with a semicircle on top, as shown. Suppose the perimeter of a particular Palladian window is 28 ft. Express the total area of the window as a function of x.

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First of all, what are you using x to represent in each of those questions?

**Aravsion** · September 17th 2011, 10:58 PM

well, in question one x would represent the width and the same with question two... I attached a picture for question two also

**Prove It** · September 17th 2011, 11:04 PM

Originally Posted by Aravsion

well, in question one x would represent the width and the same with question two... I attached a picture for question two also

There's not enough information in the first question to have the volume be a function only of x. If you're letting x represent the width, then

$\displaystyle \begin{align*} P &= 2w + 2l \\ 108 &= 2x + 2l \\ 54 &= x + l \\ l &= 54 - x \end{align*}$

As for the volume...

$\displaystyle \begin{align*} V &= lwh \\ &= (54-x)xh \\ &= (54x - x^2)h \end{align*}$

Unless you know something about the height of the box, your function will be a function of both x and h...

**Aravsion** · September 17th 2011, 11:32 PM

im just reading the problem statement from the worksheet we got in class from the professor. The first problem above we did in class but i didn't understand how and why we did each step... For example, I dont understand why do we plug in 108 into 108 = 2x + 2y... the final answer that my professor gave us is: $V = x^{2}(54-x)$

**Prove It** · September 17th 2011, 11:34 PM

Originally Posted by Aravsion

im just reading the problem statement from the worksheet we got in class from the professor. The first problem above we did in class but i didn't understand how and why we did each step... For example, I dont understand why do we plug in 108 into 108 = 2x + 2y... the final answer that my professor gave us is: $V = x^{2}(54-x)$

Maybe because you're told the perimeter is 108 inches...

**Aravsion** · September 17th 2011, 11:38 PM

i understand that the perimeter is 108 inches...... what i dont understand is the actual = 2x + 2y...

**Prove It** · September 18th 2011, 04:26 AM

Originally Posted by Aravsion

i understand that the perimeter is 108 inches...... what i dont understand is the actual = 2x + 2y...

Think of a rectangle with length = x and width = y. How do you evaluate its perimeter?

**Aravsion** · September 18th 2011, 08:53 AM

ok i understand that the perimeter is the sum of all the sides but when it comes to actually transfering English (words) into algebra that's where i have a problem... For example, when it says a maximum combined length and girth of 108" what EXACTLY does that mean? Another thing: a cross section is like cutting a figure into slices and seeing a 2-D view of it? if that is so, then when slicing a 3-D box you will solely see a square?

**skeeter** · September 18th 2011, 10:37 AM

A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108". Express the volume of such a package as a function of x.

**Prove It** · September 18th 2011, 07:02 PM

Originally Posted by skeeter

Is it safe to assume that one of the cross-sections is a square?

**Aravsion** · September 18th 2011, 07:54 PM

Prove It, well, in my personal opinion, public education in US is very weak and does not provide the very needs (the basics) for higher education.. As my college professor says forget everything that you have been taught in high school; high school is like 12 years of horror, but this is college now... I agree with him.. I mean our Precalculus teacher didn't even want to show us more examples and attempted to cram everything known in Mathematics in just half a year...

is that reasonable??
so you can say im very weak in math

as far as Palladian Window problem goes: this is how i would explain this problem to someone... we must write the area as Area = exes; thus we're not looking for a particular ANSWER we're looking how well we can manipulate equations and make subsititutions....
Formulas to consider:
Area of a circle = $\pi r^{2}$ ---> now essentially, we have half a circle [semicircle] thus we must divide by 2
Circumference of a circle = $2\pi r$ ---> now essentially, we have a half circle [semicircle] thus we must divide by 2 on both sides
-------------------------------
Because we are given the quantity of the perimeter we first must solve for it:
$x+2y+(\pi\cdot r)$
$Radius = \frac{x}{2}$ ---> this is so because we have a diameter of our circle (x) in order to have a radius we take x and divide it by 2
$x+2y+\frac{x}{2}\pi$
$2y + x(1+\frac{\pi }{2}) = 28$ ---> we factor out the x and plug in the 28 ft as our GIVEN perimeter
$2y + x(1+\frac{\pi }{2})- x(1+\frac{\pi }{2})= 28 ft - x(1+\frac{\pi }{2})$ ---> here, what we're doing is, we're taking the factored out terms and subtracting them from both sides (in other words moving/isolating the y)
$\frac{2y}{2} = \frac{28}{2} - \frac{x}{2}(1 + \frac{\pi}{2})$
$y = 14 - \frac{x}{2}(1 +\frac{\pi }{2})$ ---> we divide everything [on both sides] by 2 in order to have 1 y
So, now solving for the area
$xy+ \frac{\pi r^{2}}{2}$
$xy + \frac{1}{2}\pi (\frac{x}{2})^{2}$ ---> here, we're taking the radius and for radius we're using x/2 again and taking the formula for half of area of a circle
$x(14-\frac{x}{2}(1+\frac{\pi }{2}))+\frac{1}{2}\pi \frac{x^{2}}{4}$ ---> now we know that y = ... thus we plug in the y into the xy
$x(14-\frac{x}{2}(1+\frac{\pi }{2}))+ \frac{x^{2}\pi }{8}$
$14x - \frac{x^{2}}{2} - \frac{x^{2}\pi }{4} + \frac{x\pi }{8}$
$14x - \frac{x^{2}}{2} - \frac{x^{2}\pi }{8} = Area$

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