Respuesta :
let X and Y be the two dimensions.
Area will be X times Y.
XY = 3000000
Y = 3000000 / X
Perimeter = 2x + 3y
substituting the values:
Perimeter = 2x + 3*3000000 / x
= 2x + 9000000 / x
2 = 9000000 / x^2
x^2 = 9000000 / 2 = 4500000
x = sqrt(4,500,000)
= 2121.32
= 1000 sqrt (4.5)
y = 3000000 / 2121.32
= 1414.2
= 1000 sqrt(2)
so the dimensions will be 2121.32 by 1414.2 .
Area will be X times Y.
XY = 3000000
Y = 3000000 / X
Perimeter = 2x + 3y
substituting the values:
Perimeter = 2x + 3*3000000 / x
= 2x + 9000000 / x
2 = 9000000 / x^2
x^2 = 9000000 / 2 = 4500000
x = sqrt(4,500,000)
= 2121.32
= 1000 sqrt (4.5)
y = 3000000 / 2121.32
= 1414.2
= 1000 sqrt(2)
so the dimensions will be 2121.32 by 1414.2 .
The shortest length of fence that the rancher can use [tex]\boxed{14758{\text{ feet}}}.[/tex]
Further explanation:
Given:
The area of the rectangular field is [tex]3000000{\text{ fee}}{{\text{t}}^2}.[/tex]
Explanation:
Consider the length of the rectangular field be “x”.
Consider the width of the box as “y”.
The area of the rectangular field can be expressed as follows,
[tex]\boxed{Area=\left( x \right) \times\left( y\right)}[/tex]
The given area of the field is [tex]3000000{\text{ fee}}{{\text{t}}^2}.[/tex]
[tex]\begin{aligned}Area&= 3000000\\\left( x \right) \times \left( y \right)&= 3000000\\y&= \frac{{3000000}}{x}\\\end{aligned}[/tex]
The perimeter of the rectangular field with a fence down in the middle parallel to one side can be expressed as follows,
[tex]\begin{aligned}{\text{Perimeter}}&= 2x + 3y\\&=2x + 3\times\frac{{3000000}}{x}\\&= 2x + \frac{{9000000}}{x}\\&= \frac{{2{x^2} + 9000000}}{x}\\\end{aligned}[/tex]
Differentiate the above equation with respect to x.
[tex]\begin{aligned}\frac{d}{{dx}}\left( {{\text{Perimeter}}}\right) &= \frac{d}{{dx}}\left( {2x +\frac{{9000000}}{x}} \right)\\&=2-\frac{{9000000}}{{{x^2}}}\\\end{aligned}[/tex]
Again differentiate with respect to x.
[tex]\dfrac{{{d^2}}}{{d{x^2}}}\left( {{\text{Perimeter}}} \right)=+ \dfrac{{18000000}}{{{x^3}}}[/tex]
Substitute the first derivative equal to zero.
[tex]\begin{aligned}2- \frac{{9000000}}{{{x^2}}}&= 0\\2&= \frac{{9000000}}{{{x^2}}}\\{x^2}&= \frac{{9000000}}{2}\\{x^2}&= 45000000\\x&= \sqrt {45000000}\\x&= 6708.20\\\end{aligned}[/tex]
The value of y can be calculated as follows,
[tex]\begin{aligned}y&=\frac{{3000000}}{{6708.20}}\\&= 447.21\\\end{aligned}[/tex]
The shortest length of fence that the rancher can use [tex]\boxed{14758{\text{ feet}}}.[/tex]
Learn more:
1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.
2. Learn more about equation of circle brainly.com/question/1506955.
3. Learn more about range and domain of the function https://brainly.com/question/3412497
Answer details:
Grade: High School
Subject: Mathematics
Chapter: Area
Keywords: square, area, fence, rancher, feet, rectangle field, divide, half, fence down, middle parallel, one side, shortest length, perimeter, circumference.
