Question
Maximize Z = 50x + 30y
Subject to
$2\text{x}+\text{y}\leq18$
$3\text{x}+2\text{y}\leq34$
$\text{x},\text{y}\geq0$
Subject to
$2\text{x}+\text{y}\leq18$
$3\text{x}+2\text{y}\leq34$
$\text{x},\text{y}\geq0$
✓
Answer
First, we will convert the given inequations into equations, we obtain the following equations:
2x + y = 18, 3x + 2y = 34
Region represented by 2x + y ≥ 18:
The line 2x + y = 18 meets the coordinate axes at A(9, 0) and B(0, 18) respectively. By joining these points we obtain the line 2x + y = 18.
Clearly (0,0) does not satisfies the inequation 2x + y ≥ 18.
So,the region in xy plane which does not contain the origin represents the solution set of the inequation 2x + y ≥ 18.
Region represented by 3x + 2y ≤ 34:
The line 3x + 2y = 34 meets the coordinate axes at $\text{C}\Big(\frac{34}{3},0\Big)$ and D(0, 17) respectively. By joining these points we abtain the line 3x + 2y = 34.
Clearly (0. 0) satisfies the inequation 3x + 2y ≤ 34. So, the region containing the origin represents the silution set of the inequation 3x + 2y ≤ 34.
The corner of the feasible region are A(9, 0), $\text{C}\Big(\frac{34}{3},0\Big)$ and E(2, 14).
The values of Z at these corner points are as follows.
Therefore, the maximum value of Z $\frac{1700}{3}$ is at the point $\Big(\frac{34}{3},0\Big)$.
Hence, $\text{x}=\frac{34}{3}$ and $\text{y}=0$ is the optimal solution of the given LPP.
Thus, the optimal value of Z is $\frac{1700}{3}$.
2x + y = 18, 3x + 2y = 34
Region represented by 2x + y ≥ 18:
The line 2x + y = 18 meets the coordinate axes at A(9, 0) and B(0, 18) respectively. By joining these points we obtain the line 2x + y = 18.
Clearly (0,0) does not satisfies the inequation 2x + y ≥ 18.
So,the region in xy plane which does not contain the origin represents the solution set of the inequation 2x + y ≥ 18.
Region represented by 3x + 2y ≤ 34:
The line 3x + 2y = 34 meets the coordinate axes at $\text{C}\Big(\frac{34}{3},0\Big)$ and D(0, 17) respectively. By joining these points we abtain the line 3x + 2y = 34.
Clearly (0. 0) satisfies the inequation 3x + 2y ≤ 34. So, the region containing the origin represents the silution set of the inequation 3x + 2y ≤ 34.
The corner of the feasible region are A(9, 0), $\text{C}\Big(\frac{34}{3},0\Big)$ and E(2, 14).
The values of Z at these corner points are as follows.
|
$\text{Corner point}$
|
$\text{Z}=50\text{x}+30\text{y}$
|
|
$\text{A}(9, 0)$
|
$50\times9+3\times0=450$
|
|
$\text{C}\Big(\frac{34}{3},0\Big)$
|
$50\times\frac{34}{3}+30\times0=\frac{1700}{3}$
|
|
$\text{E}(2, 14)$
|
$50\times2+30\times14=520$
|
Hence, $\text{x}=\frac{34}{3}$ and $\text{y}=0$ is the optimal solution of the given LPP.
Thus, the optimal value of Z is $\frac{1700}{3}$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{e}^{3\text{x}}\sin4\text{x}\times2^\text{x}$
→$\text{y}=\text{e}^{3\text{x}}\sin4\text{x}\times2^\text{x}$
Using definite intergeals, find the area of the region bounded by the following curves, after making a rough sketch y = 1 + |x + 1|, x = -2, y = 0.
Solve the following initial value problems:
$\text{xe}^{\frac{\text{y}}{\text{x}}}-\text{y + x}\frac{\text{dy}}{\text{dx}}=0,\text{y(e)}=0$
→$\text{xe}^{\frac{\text{y}}{\text{x}}}-\text{y + x}\frac{\text{dy}}{\text{dx}}=0,\text{y(e)}=0$
Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}2&3&7\\13&17&5\\15&20&12 \end{vmatrix}$
→$\begin{vmatrix}2&3&7\\13&17&5\\15&20&12 \end{vmatrix}$
Solve the following systems of homogeneous linear equations by matrix method:
→$x + y - z = 0$
$x - 2y + z = 0$
$3x + 6y - 5z = 0$
Write the set of values of 'a' for which $\text{f}(\text{x})=\log_\text{a}\text{x}$ is increasing in its domain.
→Prove that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is $\frac{2\text{R}}{\sqrt3}$ Also finds the maximum volume.
→If x = a ( θ – sin θ ), y = a (1 + cos θ ), find $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}$.
→The total area of a page is $150cm^2$. The combined width of the margin at the top and bottom is $3cm$ and the side $2cm$. What must be the dimensions of the page in order that the area of the printed matter may be maximum?
→Find the equatoion of the passing through the points (1, -1, 2) and (2, -2, 2) and which is perpendicular to the plane 6x - 2y + 2z = 9.