Question

solve using differentiation:

a triangle has a base 12 feet long and an altitude 8 feet high. find the area of the largest rectangle that can be inscribed in the triangle so that the base of the rectangle falls on the base of the triangle.
answer is 24 sq. ft.
*needs solution

Answer 1

@uri

Answer 2

is it a right triangle? an equilateral triangle, or just some triangle?

Answer 3

given the base and the altitude as 12 ft and 8 feet respectively,

Answer 4

so that the base of the rectangle falls to the base of triangle. How would be the figure of that..

Answer 5

the point \(y\) labelled on the right satisfies the equation \(y=-3x+12\) the length of the base of the rectangle is \(2x\) and so the area of the rectangle is
\[A(x)=2x(-3x+12)=-6x^2+24x\]

Answer 6

you don't need calculus to find the max of that quadratic
find the first coordinate of the vertex
that will give you \(x\)

Answer 7

so how can we come up with 24 sq ft? sorry i'm really not good at this. :(

Answer 8

first of all is it clear how i came up with \(y=-3x+12\) ?

Answer 9

Because of the equation of the side?

Answer 10

to answer your question, how do you come up with \(24\) the vertex of the quadratic,
\[A(x)=-6x^2+24x\] is \((2,24)\) and there is the 24

if you want to use calculus to find the vertex, take the derivative of \(A(x)=-6x^2+24\) and get \(A'(x)=-12x+24\)
set it equal to zero and solve, and get \(x=2\)
then you find that \(A(2)=24\)

Answer 11

yes, the equation of the line through \((0,12)\) and \((4, 0)\) is \(y=-3x+12\)

Answer 12

the real key here is to stick the triangle inside a coordinate axis in order to get an equation for the area
without that, it is almost impossible
once you use a coordinate axis, the equation for the area is not that hard to find

Answer 13

hm.. i think i get it already. wait i'll just going to solve for it on my own. thankyou for the help.

Answer 14

can i ask you another problem later?

Answer 15

yw
sure you can ask later

Answer 16

Thank you. :)

Answer 17

can you show me the solution to this y=−3x+12.? i think i'm getting it wrong -_-

Answer 18

you can do that in your head
the line from \((0,12)\) to \((4,0)\) has \(y\) intercept 12 right?

Answer 19

as for the slope, you go over 4, down 12, so slope is \(-\frac{12}{4}=-3\)

Answer 20

like with slope \(-3\) and \(y\) intercept \(12\) is \(y=-3x+12\)

Answer 21

i have another solution ..\[y=-\frac{ 12 }{ 8 }x +12\] would give me a x=4
and then substituting it to our equation of the line give me y=6
Area of rec = xy = (4)(6)=24