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Solving a math word problem (quadratic formulas)?
Sylvia has an apple orchard. One season, her 100 trees yielded 140 apples per tree. She wants to increase her production by adding more trees to the orchard. However, she knows that for every 10 additional trees she plants, she will lose 4 apples per tree (i.e., the yield per tree will decrease by 4 apples). How many trees should she have in the orchard to maximize her production of apples?
I've tried all sorts of approaches but can't figure it out... I know the answer is 225 but I do not know how to reach it. I assumed I need 3 equations but that gives me 3 unknowns and I tried the elimination method unsuccessfully. I think I'm making this problem a lot harder than it should be... Any help is much appreciated!!
2 Answers
- Jeff AaronLv 71 decade agoFavorite Answer
Suppose she plants t trees, and each one yields a apples per tree, therefore:
a = 140 - 4((t - 100)/10)
a = 140 - 0.4t + 40
a = 180 - 0.4t
If y is the total yield for the orchard, we have:
y = ta
y = t(180 - 0.4t)
y = 180t - 0.4t^2
dy/dt = 180 - 0.8t
To maximize y, set the derivative to zero:
0 = 180 - 0.8t
0.8t = 180
t = 180 / 0.8
t = 225
- 4 years ago
set up your 2 cases (keep in mind that the gap wherein the automobile travels keeps to be consistent in spite of the shown fact that its velocity and time differences). ---------------- Case a million) typical velocity velocity = x Distance = one hundred and five Time = y Case 2) larger velocity velocity = x + 10 Distance = one hundred and five Time = y - a million/4 OR (4y - a million)/4 Now type your 2 simultaneous equations (applying velocity = distance/time): x = one hundred and five/y x + 10 = one hundred and five / (4y - a million)/4 Simplify the complicated fraction: x + 10 = 420 / (4y - a million) ------------------- Now exchange one hundred and five/y for x interior the 2nd equation: one hundred and five/y + 10 = 420 / (4y - a million) Multiply each and every thing by ability of y(4y - a million) to cancel off all denominators: one hundred and five(4y - a million) + 10y(4y - a million) = 420y Divide each and every thing by ability of 5 to simplify the equation: 21(4y - a million) + 2y(4y - a million) = 84y amplify all the brackets: 84y - 21 + 8y^2 - 2y = 84y Subtract 84y from the two factors and type from optimum ability to lowest: 8y^2 - 2y - 21 = 0 element it out: (2y + 3)(4y - 7) = 0 this can furnish you the possibility of the time the automobile travelled in being the two -3/2 hours or 7/4 hours. Now of course you may no longer return and forth lower back in time so the time wherein the automobile travelled can in undemanding terms be 7/4 hours. Now which you comprehend what the time is stumble on that to discover the fee (applying velocity = distance/time): x = one hundred and five / (7/4) x = one hundred and five * 4/7 x = 60 the unique velocity of the automobile replaced into 60mph.
