Hey there!!
Let us take the price of the redwood as ' x ' and pine as ' y '
Then , we take it into an equation. We get,
50x + 80y = 285 -------------- ( 1 )
80x + 50y = 339 -------------- ( 2 )
Now, multiply the first equation with 8 and the second equation with 5
400x + 640y = 2280
400x + 250y = 1695
Now subtract the second equation from the first
390y = 585
y = 585 / 390
y = $1.5
Now substitute this into any equation
50x + ( 80 ) ( 1.5 ) = 285
50x + 120 = 285
50x = 165
x = 165 / 50
x = $3.3
Redwood = $3.3
Pine = $1.5
Hope helps!
Split up the interval [0, 2] into <em>n</em> equally spaced subintervals:
![\left[0,\dfrac2n\right],\left[\dfrac2n,\dfrac4n\right],\left[\dfrac4n,\dfrac6n\right],\ldots,\left[\dfrac{2(n-1)}n,2\right]](https://tex.z-dn.net/?f=%5Cleft%5B0%2C%5Cdfrac2n%5Cright%5D%2C%5Cleft%5B%5Cdfrac2n%2C%5Cdfrac4n%5Cright%5D%2C%5Cleft%5B%5Cdfrac4n%2C%5Cdfrac6n%5Cright%5D%2C%5Cldots%2C%5Cleft%5B%5Cdfrac%7B2%28n-1%29%7Dn%2C2%5Cright%5D)
Let's use the right endpoints as our sampling points; they are given by the arithmetic sequence,

where
. Each interval has length
.
At these sampling points, the function takes on values of

We approximate the integral with the Riemann sum:

Recall that

so that the sum reduces to

Take the limit as <em>n</em> approaches infinity, and the Riemann sum converges to the value of the integral:

Just to check:

{ 3-y=2x
{ x+1/2y=3/2
{ -y=2x-3
{ 1/2y=3/2-x
{ y=-2x+3
{ y=3-2x
-2x+3=3-2x
(x,y) = (x,-2x+3)
Nineteen and two hundred and thirty eight thousandths