hawkfalcon
  • hawkfalcon
The height of triangle ACE is 12 and base AE = 20. A rectangle is inscribed in triangle ACE as shown below.? a) Express the rectangle’s area as a function of x and y. Use the fact that the triangle ACE is similar to triangle BCD to write y as a function of x. c)Express the rectangle’s area as a function of only x. d) Find the maximum possible area of the rectangle.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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hawkfalcon
  • hawkfalcon
|dw:1345995812545:dw|
hawkfalcon
  • hawkfalcon
Nobody knows what to do?:#
hawkfalcon
  • hawkfalcon
@Zekarias ?

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More answers

anonymous
  • anonymous
a) (12-y)x b) 12(x-0.1x^2) c) 30 sq units
hawkfalcon
  • hawkfalcon
...How?
anonymous
  • anonymous
Is "How" for all of it?
hawkfalcon
  • hawkfalcon
Just b and c
hawkfalcon
  • hawkfalcon
especially b... :O
anonymous
  • anonymous
consider the perpendicular line form C, we can come out with the ratio 12/y =10/x, where 10 is half the rectangles side,20. Then simplify it so as to inset the result in to the equation in 'a'. To 'c' just use the application of derivative.
hawkfalcon
  • hawkfalcon
Okay. Makes sense I guess:p thanks!
ganeshie8
  • ganeshie8
small catch
ganeshie8
  • ganeshie8
triangle ACE is similar to triangle BCD => height1/height2 = base1/base2
ganeshie8
  • ganeshie8
12/y = 20/x y = (3/5)x
ganeshie8
  • ganeshie8
this would give you max area 60
hawkfalcon
  • hawkfalcon
o.O But y needs to be written as a function of x.
ganeshie8
  • ganeshie8
y = (3/5)x
ganeshie8
  • ganeshie8
its a function of x
ganeshie8
  • ganeshie8
linear
hawkfalcon
  • hawkfalcon
Oh yeah :3. So why did you use the entire base and @Zekarias used 1/2?
ganeshie8
  • ganeshie8
since both are similar triangles, "ratio of bases" equals "ratio of heights"
hawkfalcon
  • hawkfalcon
Okay, that makes sense. :3 Can you explain how you got 60 though?
ganeshie8
  • ganeshie8
write the area equation in x and optimize
ganeshie8
  • ganeshie8
for optimizing, use either of the below two : 1) its a parabola facing down so max value = f(-b/2a) or 2) use calculus
ganeshie8
  • ganeshie8
this q from conics, or calculus ?
hawkfalcon
  • hawkfalcon
calculus. Going into calculus next year and they gave us some prep stuff*.*
ganeshie8
  • ganeshie8
oh ok, so we have this so far : A = x(12-y) -------(1) y = (3/5)x --------(2)
hawkfalcon
  • hawkfalcon
Okay, true.
ganeshie8
  • ganeshie8
hmm which one moves ?
ganeshie8
  • ganeshie8
ohhk yeah as we go down its moving a lot it seems...
ganeshie8
  • ganeshie8
substitute (2) in (1)
hawkfalcon
  • hawkfalcon
x(12-(3/5)x)
ganeshie8
  • ganeshie8
\(A = x(12 - \frac{3}{5}x)\) \(A = 12x- \frac{3}{5}x^2)\)
hawkfalcon
  • hawkfalcon
Okayy and solve for x
ganeshie8
  • ganeshie8
nope we need to optimize A
ganeshie8
  • ganeshie8
find maximum possible value of A, right ?
hawkfalcon
  • hawkfalcon
Oh.
hawkfalcon
  • hawkfalcon
Ya
ganeshie8
  • ganeshie8
theres a process for finding maximum/minimum values of any funcitons
ganeshie8
  • ganeshie8
you familiar with differentiation ?
hawkfalcon
  • hawkfalcon
... I guess I havenet learned that yet..
ganeshie8
  • ganeshie8
its simple... if you knw differentiation already
hawkfalcon
  • hawkfalcon
I think we are learning that next year :( (monday)
ganeshie8
  • ganeshie8
just if you knw derivative of x, and x^2, its enough
ganeshie8
  • ganeshie8
oh ok then you cannot use calculus, lets switch to conics. you knw parabolas, right
hawkfalcon
  • hawkfalcon
Okay, I don't know how to do that.. Maybe I should make a new question asking for the dirivative of A=12x-3/4x^2
hawkfalcon
  • hawkfalcon
Yes.
hawkfalcon
  • hawkfalcon
You said f(-b/2a)
ganeshie8
  • ganeshie8
yes
ganeshie8
  • ganeshie8
A = 12x - (3/5) x^2
ganeshie8
  • ganeshie8
\(A = \frac{-3}{5}x^2 + 12x\)
hawkfalcon
  • hawkfalcon
Yes
ganeshie8
  • ganeshie8
a = -3/5 b = 12
hawkfalcon
  • hawkfalcon
Cool, thanks, just plug and chug now:D
ganeshie8
  • ganeshie8
f(-b/2a) gives the vertex y co-ordinate, which is the maximum value
hawkfalcon
  • hawkfalcon
y=10
ganeshie8
  • ganeshie8
|dw:1345999525420:dw|
ganeshie8
  • ganeshie8
-b/2a = 10 f(-b/2a) = ?
ganeshie8
  • ganeshie8
put x = -b/2a = 10, in area function
hawkfalcon
  • hawkfalcon
Okay.
hawkfalcon
  • hawkfalcon
60!:D
ganeshie8
  • ganeshie8
if you remember, for any parabola, -b/2a is the x-coordinate of vertex f(-b/2a) is the y-coordinate of vertex
ganeshie8
  • ganeshie8
great !!
hawkfalcon
  • hawkfalcon
Thank you so much! Now I can do the rest of these problems:D
ganeshie8
  • ganeshie8
glad to hear ! calculus wud be much fun :) good luck !!

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