There are 50 apple trees in an orchard. Each | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let $x$ be the number of additional trees planted.The total number of trees becomes $(50 + x)$.The production per tree becomes $(800 - 10x)$.The t

Vedclass · Accepted Answer

$15$

Vedclass · Answer

(C) Let $x$ be the number of additional trees planted.The total number of trees becomes $(50 + x)$.The production per tree becomes $(800 - 10x)$.The total production $P(x)$ is given by the product of the number of trees and the production per tree:$P(x) = (50 + x)(800 - 10x)$$P(x) = 40000 - 500x + 800x - 10x^2$$P(x) = -10x^2 + 300x + 40000$To maximize the output,we find the derivative $P'(x)$ and set it to $0$:$P'(x) = \frac{d}{dx}(-10x^2 + 300x + 40000) = -20x + 300$Setting $P'(x) = 0$:$-20x + 300 = 0$$20x = 300$$x = 15$To verify this is a maximum,we check the second derivative:$P''(x) = -20$Since $P''(x) < 0$,the function has a maximum at $x = 15$.

There are $50$ apple trees in an orchard. Each tree produces $800$ apples. For each additional tree planted in the orchard,the output per tree drops by $10$ apples. The number of trees that should be added to the existing orchard to maximize the total output is:

Similar Questions

The denominator of a fraction is $16$ more than the square of the numerator. Find the least value of the fraction.

The abscissae of the points,where the tangent to the curve $y = x^3 - 3x^2 - 9x + 5$ is parallel to the $x$-axis,are

An open tank with a square bottom is to contain $4000 \ cm^3$ of liquid. Find the dimensions of the tank such that the surface area of the tank is minimum.

If $m$ and $M$ respectively denote the minimum and maximum of $f(x)=(x-1)^2+3$ for $x \in [-3, 1]$, then the ordered pair $(m, M)$ is equal to

If a running track of $500 \ ft$. is to be laid out enclosing a playground,the shape of which is a rectangle with a semicircle at each end,then the length of the rectangular portion such that the area of the rectangular portion is to be maximum is (in feet).

Vedclass Test Series

Exam Paper Generator

Online Exam Module