My Math Forum One question's stumping me

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 October 1st, 2019, 10:47 AM #1 Senior Member   Joined: Mar 2019 From: TTF area Posts: 168 Thanks: 1 One question's stumping me Tom wants to sell some land (the diagram below). Before he can sell the land, he needs to install a pipe which costs 50 dollars per meter. A property developer is only interested if the block of land can be subdivided into four sections of at least 700 square meters (700 m^2) each. The straight pipe goes from point A to C (shown as - - - - - on the diagram). This pipe must be one of the boundaries for all four sections so that each section can have easy access to this pipe. the activity requires you to provide necessary calculations and write a report to determine whether the subdivision meets the property developer’s needs. The land for sale is bounded by ABCD and needs to be split into four sections. The pipe running through the land is between points A and C (shown as - - - - - ) and each of the four sections must-have part of AC as one of its boundaries. 1. Calculate the length and cost of the pipe going through the land from point A to point C. I calculated this to be 62.45m, so the cost would be 3122.5 dollars 2. Demonstrate that the land can be divided into four sections each of more than 700 m2 in order to meet the buyer’s requirements for the sale to proceed. Area of ABC=1462 approx and Area of ACD=1515 approx so that = 2977m^2 more than 4x700m^2 so there can be four sections of more than 700m^2. This is the one I'm stuck at I've tried a lot and haven't had any luck. 3. The buyer has also decided that he will not purchase the land if all of the sections are triangular. Produce a report justifying the sale of the land showing one possible way of dividing the land into four sections each of more than 700 m^2 . Show the dimensions of each of the sections. Last edited by skipjack; October 1st, 2019 at 11:42 AM.
 October 1st, 2019, 11:36 AM #2 Global Moderator   Joined: Dec 2006 Posts: 21,035 Thanks: 2271 Can you provide the diagram?
October 1st, 2019, 12:17 PM   #3
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Yeah, I thought I did sorry, I accidentally posted it twice and the diagram must have been on the other one.
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 Screenshot 2019-09-26 at 12.27.29 PM.png (15.8 KB, 7 views) Screenshot 2019-10-02 at 7.42.21 AM.png (7.1 KB, 1 views) Screenshot 2019-10-02 at 7.41.41 AM.png (7.2 KB, 0 views)

October 1st, 2019, 02:34 PM   #4
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Divided triangle ADC into two regions of approximately equal area, a triangle and a trapezoid, each with area > 700 m^2 (see attached diagram). Used the concept of the square of side ratios to area ratios for similar triangles EDF & ADC.

You can do the same for triangle ABC which will, in total, yield two triangles and two trapezoids (not all triangles as desired by the buyer).
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 area_prob2.jpg (67.3 KB, 4 views)

 October 1st, 2019, 04:40 PM #5 Senior Member   Joined: Mar 2019 From: TTF area Posts: 168 Thanks: 1 That's what I was thinking but the question states that: each of the four sections must-have part of AC as one of its boundaries. So this is why I'm confused
 October 1st, 2019, 04:40 PM #6 Senior Member   Joined: Jun 2019 From: USA Posts: 310 Thanks: 162 Skeeter's answer does not satisfy each lot bordering the pipe. You need a property line going from the pipe to a point on AB or CB, and one going from the pipe to AD or CD. No real constraints, but to make calculations easy you might let EF be a property line perpendicular to AC such that triangle CEF and quadrilateral ADFE have the same area. Then make a line GH perpendicular to AC such that triangle CGH and quadrilateral ABHG have the same area. If ABC and ACD are both at least 1400 m², I'm pretty sure this method will work. (Possibly with some narrow plots, but heck, the buyer's constraint was rather odd to begin with.) Thanks from helpmeddddd
October 1st, 2019, 05:09 PM   #7
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Right, I've done some maths and this is what I got. if you draw a line from the middle of the pipe AC (31.225( to the corner of AB (angle 56.081) I get a triangle in the corner with an area of about 733m^2 and the area of the non triangle must then for be 1462-733= 729m^2 so that works nicely . For ACD triangle, I again drew a line from the middle of line AC to the corner of AD (angle 60). I got that triangle's area to be 757.8m^2 and the other triangle's area was 758m^2, added them up and got about 1515. which is right!!!

I'll attach an image - not an accurate one, but it gives you some idea what I'm going on about!!
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 Screenshot 2019-10-02 at 2.08.22 PM.jpg (19.3 KB, 4 views)

Last edited by skipjack; October 2nd, 2019 at 08:55 PM.

October 1st, 2019, 05:18 PM   #8
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oops ... sorry about the omission of the given condition

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 area_prob3.jpg (30.7 KB, 3 views)

Last edited by skeeter; October 1st, 2019 at 05:24 PM.

 October 2nd, 2019, 11:11 AM #9 Senior Member   Joined: Mar 2019 From: TTF area Posts: 168 Thanks: 1 Yeah, skeeter, that's more or less what I did. @DarnitjimimAnEnigeneer is my post #7 correct? Last edited by skipjack; October 2nd, 2019 at 08:55 PM.
 October 2nd, 2019, 11:39 AM #10 Senior Member   Joined: Jun 2019 From: USA Posts: 310 Thanks: 162 I didn't check the numbers. It looks about right, though.

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