Question

A farmer has 70-acres on which to plant a pear orchard. Three neighboring farms with similar soil conditions already have established orchards. One of these orchards has 250 trees planted per acre yielding, on average, 525 pears per tree. Another has 280 trees per acre, yielding 480 pears per tree, while the third with 350 trees per acre yields only 375 pears per tree (see chart).

Pear Orchard Crop Production 600 250,525 280,480 400 Yield (pears/tree) 350,375 200 0 0 100 200 300 400 Density (trees/acre)

The average weight of a pear is approximately ½-pound. Perform appropriate transformations so as to express the total yield from the farmer’s 70-acres, measured in tons (rather than pears), as a function of the number of trees per acre. What is the anticipated maximum yield (in tons) from the farmer’s 70-acre orchard?Based on this data, derive equations for both the yield per tree and the yield per acre as functions of x, the tree density (trees/acre). What density of planting is expected to yield no pears? What density does your model suggest will produce the maximum yield per acre?

1. Let T(x) represent the yield in pears per tree when x trees are planted per acre. Based on the given chart this relationship appears to be linear. Using any two of the data points, calculate the slope of the graph of T(x). ??

2. Write an equation for T(x), the yield in pears per tree, as a linear function in slope-intercept form.

The equation is T(x) =??x +??

3. Identify the y-intercept of the graph of T(x).

4. Identify the x-intercept of the graph of T(x).

5. Considering contextual restrictions on both trees per acre and the yield per tree, which of the following is the most appropriate domain for the function T(x)?

(a) (0,∞)

(b) [0, 900)

(c) (-∞,∞)

(d) (0, 600]

6. Considering contextual restrictions on both trees per acre and yield per tree, which of the following is the most appropriate range for the function T(x)?

(a) [0, 900)

(b) (-∞,∞)

(c) (0, 600]

(d) (0, ∞)

7. The yield in pears per acre can be expressed as a quadratic function in the standard form A\left(x\right)=ax^2+bx+cA(x)=ax2+bx+c. Identify the values for the coefficients a, b and c.

Note: a doesn't = 0, but either b or c may equal zero.

a =??;b =??;c =??

8. Every quadratic function is either concave up or concave down and has either an absolute maximum or absolute minimum. Based solely on the equation for A(x) select the statement that best describes this function.

(a )A(x) has a positive leading coefficient, meaning its graph is concave down and it has an absolute minimum.

(b) A(x) has a positive leading coefficient, meaning its graph is concave up and it has an absolute minimum.

(c)A(x) has a negative leading coefficient, meaning its graph is concave down and it has an absolute maximum.

(d)A(x) has a negative leading coefficient, meaning its graph is concave up and it has an absolute maximum.

9. Identify the vertex of A(x).

The vertex is the point(??,??)

10. What is the optimal number of trees to plant per acre in order to produce the maximum yield per acre?

11. At the optimal tree density what is the total projected yield (in pears) from the farmer's 70-acre orchard?

12. Considering contextual restrictions on both trees per acre and the yield per acre, identify an appropriate domain for A(x).

(a) [0, 600]

(b) [0, 135000]

(c) (-∞,∞)

(d) [0,∞)

(e) [0, 300]

13.

Considering contextual restrictions on both trees per acre and the yield per tree, identify the most appropriate range of the function T(x). .

[??,??]

14. Identify the y-intercept of A(x).

15. Identify any x-intercepts of the graph of A(x). Select all the apply:

(a) (300,0)

(b) (300,135000)

(c) (0,300)

(d) (0,600)

(e) (600,0)

(f) (0,0)

16. Complete the equation for the axis of symmetry of A(x).

??=??

17. Complete the following statement:

Keeping in mind contextual restrictions on the domain, the function A(x) is increasing over the open interval(??,??)and decreasing over the open interval(??,??).

18. Select the correct equation for Y(x), the total projected yield in tons from the farmer's 70-acre orchard.

(a) y(x)=-0.105x^2+63x

(b) y(x)=-0.02625x^2+15.75x

(c) y(x)=-105000x^2+63000000x

(d) y(x)=-0.000375x^2+0.225x

Answer 1

Solution yield (pears per 1250,525) (280,480) * (850,375) free) Density (free facre) clearly it is a straight line eam: Y-Y = (y2 - 4 | (8-11) ( ) (2-2) tale can use any 2 point. yo yon ) (•*.) y.525+ ( (8-250) > 8 480-525 280-250 tree 2525 -115 (1-250) Tla) yout= -1158 +900 eye yield per tree density (tree /acre) for no pears , y = 0, 2) 43960 600 trees facere for yield per acre you need to multiply pear or yield per tree and number of trees (density) per acre. Scanned by CamScanner

(280,134400) (350,13.12.50 yield (pears/acre) (250,131250) Density (trees facre) When density 250 r yield per aere = 250*525 131250 280, acre - 280x480 , 134400 yield pera deusity - yieldper density. 350, acre - 350x375 - 131250 clearly the graph is parabolic as shown above eqn: y = ara²+bx+c Using all 3 points you will get 3 equations which can be solved to get a,b,c. On solung a-1.5 b:900 C=0 yo -1.50² +900x Ye yield &pears facere) deusity (freesfacere) The maximum value will be at peak where -1.5x2 x + 90070 ) 2 = 300 da dyo 7 Scanned by CamScanner

acre ... 2 for to acre 5x10 ton So yield is maximum when there is 300 trees per acre density y= -1.58² +9000 y yield in pears per xx trees facre density weight of pears. £ 16 weight of pears per acre weight of pears: 12 * 70xy, 3sy 1 pound .. weight of pears in tons: 35x5x10 ty = 0.01754 let y'= weight of pears (tons) & Outsy У? 0.975ya 0.0175 (-1.59² +900x) » 19': - 002 153* + 151 y'a total yield in 70 acre land in tons trees / acre density Max. yield occurred at 8 = 30 now, -1.58² +9001 da Scanned by CamScanner

... + 151 , x 30 Мочу - - 0-02 Сих зо? 2 з 2: tons

1) We have derived The). Thay ay () = -1.544900 lelope derivation 4 slope -1.5 ما - Slope - Yz-y, 2₂-24 : 480-525 280-250 -1.5 (3) TIM): -1.5%+ 900 (3) intercept 900 20 TCM)3900] (4) d-intercept When Tlx = 0 =) -1.5x + 900-0 X2 600 (5) plintercept: 800 Clearly beyond 28600, To: -Ve domain & (0,600] (d) [not possible To 900, da Goo Too (6) for x70, Range of Ita)= [0,900) . Scanned by CamScanner

egn we earlier derived this A(x)= Yo -1.582 +9002 las -1.5 b=900, १) We have already plotted the graph of Ala) and also calculated is maximum point. clearly we can see that it is concave down as its leading coefficient 'a' is-ve (-15). and thus it has an absolute maxima. so option (c) Best describes A(n). 9) vertee of Aca) is point of maximum. We know ye -1.582 +90on is maximunt yo 2-1:5* 300? +900*300 (verteen ( 300, 135000)] that at 2300 135000 Scanned by CamScanner

10) We already found this. acre Optimal number trees of = 300 trees/ac trees !!) The vertex denotes the point of optimal number of pears per got ye pears per acre facre and maximum yield of acre. we 135000 when a = 300. number of acres 70 .: Total yield: 70x13500o - 9450000 Verten ( 300, 135000) pears 12) + yield (pearifan) Arya -1.50 ²+9008 deusity (frees facre) Scanned by CamScanner

ale can see that a should die between 24 and a where in both conditions yoo 02 -1:5*?+ 9oom 0,600 -> 3 0, 1, Cho : domain of Ala) [0,600] 13 2 A (a) minimum [yield can't be awe A(x) maximum Range of A(M)= [0, 135000] 135000 ... 14) clearly A(x) intersects yaxis at 24 50 to yao intercept .. 15) A(&) intersects xaxis at a, =0, x=600 yo yoo A 10.0) f D(600,0) are interceps 16) Equation of anis of symmetry Parabola is symmetric about a vertical line Scanned by CamScanner

passing through its verken V line of symmetry 21 300 21 310 dy yo It) for increasing for decreasing dy co dri dy -3x +900 -> oto ya -1.5** + 9001 da clearly for xit(0,300) dy o (increasing) f for at (8300,600) * - (300 , 0o) đi < 0 (Attra-11) This can be seen from graph also. Increasing =) (0,300) decreasing - 18) We have already solved this as as y'la) M(X)y' (a) >=0.0 2625*** 15.75* D [300,600) Scanned by CamScanner