Year 12 Maths B problem solving

[unknown12]

Homework Statement

http://s1183.photobucket.com/albums/x477/mathsscholar/
This photo just show equations which define the boundaries, note there is no eastern boundary but the domain is 0-20.
After further discussions with his accountant, Mr X is not satisfied that he will make enough profit on this land development using these boundaries. He believes that he can increase his profit by cutting each block of land into two (2) smaller blocks of equal area. In order to do this, he plans to fence another straight boundary EF that is parallel to the fence BC.
The task is to find the length of the new fence EF which will cut the block of land exactly in half.

Homework Equations

DC, 0.001x^6 - 0.051x^5 + 0.87x^4 - 5.33x^3 + 5.529x^2 + 11.781x + 312, DA, -308x + 312, AB, y = 17ln(x) + 4. the area under the curve DCwith the domain is 5478m^2, I'm still yet to subtract the area under AB though, this could then be divided by 2 for half the area, what's left is to locate where the boundary would be dividing the block in half. The question states that an algebraic method should be used for an A grade.

The Attempt at a Solution

All I can display is a graph, attempting to describe the question. http://s1183.photobucket.com/albums/x477/mathsscholar/
Alright, I worked out the area under AB the total area of land is 5478-771.54897, the total area is 4706.45103m^2. Therefore the two blocks have to be 2353.225515m^2, what's left is to findthe x coordinates of the new fence. Thank you.

 

[LCKurtz]

What about the area of land above the straight line segment on the left? Where x is between 0 and 1. Have you included that? Given your numbers, I don't think so...

 

[unknown12]

What about the area of land above the straight line segment on the left? Where x is between 0 and 1. Have you included that? Given your numbers, I don't think so...

Thanks for noticing, I thought I originally had, but it seems I did miss something, I just check other working and some of it is a bit off, thanks. The new area is: 5797.3429-158-771.54897 = 4867.79393, therefore the blocks will have an area of 2433.896965 m^2. Thanks for making me check:)

 

[LCKurtz]

Thanks for noticing, I thought I originally had, but it seems I did miss something, I just check other working and some of it is a bit off, thanks. The new area is: 5797.3429-158-771.54897 = 4867.79393, therefore the blocks will have an area of 2433.896965 m^2. Thanks for making me check:)

Your numbers look correct now (at least close enough for government work ). I'm curious what "algebraic method", if that's what it is, you are using to get these numbers.

 

[unknown12]

Your numbers look correct now (at least close enough for government work ). I'm curious what "algebraic method", if that's what it is, you are using to get these numbers.

I really need to work on explaining myself better, I provided the results directly from my calculator for these numbers, I obtained similar figures using the trapezoidal rule (every .5m along x) but I just posted the more precise figures as we were to compare the results from a calculator and the results we found ourselves. Anyway, could you offer your assistance locating the x coordinate of the new fence?

 

[LCKurtz]

I really need to work on explaining myself better, I provided the results directly from my calculator for these numbers, I obtained similar figures using the trapezoidal rule (every .5m along x) but I just posted the more precise figures as we were to compare the results from a calculator and the results we found ourselves. Anyway, could you offer your assistance locating the x coordinate of the new fence?

If you call the x you are looking for x₀, you need to set the integral from 0 to x₀ of f(x) - g(x) equal to half the area, and treat x₀ as the unknown.

 

[unknown12]

If you call the x you are looking for x₀, you need to set the integral from 0 to x₀ of f(x) - g(x) equal to half the area, and treat x₀ as the unknown.

Thanks heaps! I'll put it to work tonight or tomorrow, I'll get back to you if i have any problems, once again thanks.

 

[unknown12]

If you call the x you are looking for x₀, you need to set the integral from 0 to x₀ of f(x) - g(x) equal to half the area, and treat x₀ as the unknown.

I think what you were describing to me is right, however, the results I have obtained can't be right. I integrated f(x) (let that be the top function), g(x) let that be the linear line to the left, and h(x) let that be the logarithmic. I then subtracted these values from f(x, after that I entered the function on my calculator and entered the area into Y<sub>2, the answer I obtained was 3.99, perhaps a little too small to divide the land into equal areas?I'm not familiar with logarithmic integrating but I obtained an answer of 8.5ln(x²)+4x from the original equation 17ln(x)+4. Would this method work, or is it just an integrating error, logarithmic or the polynomial.

edit: I understand what you were trying to say above but when I sub 20 into the integral, I get 67000 and to my knowledge I don't think I integrated it wrong(0.001/7x^7 - 0.051/6x^6 + 0.87/5x^5 - 5.33/4x^4 + 5.529/3x^3 + 11.781/2x^2 + 312x)-((-308/2)x^2/312x)-(8.5ln(x^2)+4x)</sub>

 

[LCKurtz]

I think what you were describing to me is right, however, the results I have obtained can't be right. I integrated f(x) (let that be the top function), g(x) let that be the linear line to the left, and h(x) let that be the logarithmic. I then subtracted these values from f(x, after that I entered the function on my calculator and entered the area into Y<sub>2, the answer I obtained was 3.99, perhaps a little too small to divide the land into equal areas?I'm not familiar with logarithmic integrating but I obtained an answer of 8.5ln(x²)+4x from the original equation 17ln(x)+4. Would this method work, or is it just an integrating error, logarithmic or the polynomial.

edit: I understand what you were trying to say above but when I sub 20 into the integral, I get 67000 and to my knowledge I don't think I integrated it wrong(0.001/7x^7 - 0.051/6x^6 + 0.87/5x^5 - 5.33/4x^4 + 5.529/3x^3 + 11.781/2x^2 + 312x)-((-308/2)x^2/312x)-(8.5ln(x^2)+4x)</sub>

<sub>For one thing, you need to check your integration of 17ln(x) + 4. Ln(x) requires integration by parts and you should have two terms from it in addition to integrating the 4.

Your final answer for x0 should be between 11 and 12.</sub>