Question
Maximum $Z = 3x + 5y$
Subject to
$\text{x}+2\text{y}\leq20$
$\text{x}+\text{y}\leq15$
$\text{y}\leq5$
$\text{x},\text{y}\geq0$
Subject to
$\text{x}+2\text{y}\leq20$
$\text{x}+\text{y}\leq15$
$\text{y}\leq5$
$\text{x},\text{y}\geq0$
✓
Answer
Converting the given inequations into equation:-
$x + 2y = 20, x + y = 15, x = y = 0$
Region represented by $x + 2y = 20:$
Line x + 2y = 20 meets coordinate axes at $A_1(20, 0)$ and $B_1(0, 10)$, clearly, (0, 0) satisfies $\text{x}+2\text{y}\leq20$, so region containing origin represents $\text{x}+2\text{y}\leq20$ in xy -plane.
Region represented by $\text{x}+\text{y}\leq15$:
Line x + y = 15 meets coordinate axes at $A_2(15, 0)$ and $B_2(0, 15)$, clearly, (0, 0) satisfies $\text{x}+\text{y}\leq15$, so region containing origin represents x + y = 15 in xy-plane.
Region represented by $\text{y}\leq5$:
Line y = 5 is parallel to x-axis and meets at $B_3(0, 5)$ on y-axis.
Clearly (0, 0) satisfies $\text{y}\leq5$, so region containing origin represents y s 5 in xy-plane.
Region represented by $\text{x},\text{y}\geq0$:
It represent the first quadrant in xy-plane.
So, shaded region$ OA_2PB_3$ represents the feasible region.
Coordinate of P(10, 5) is obtained by solving $x + 2y = 20$ and $y - 5$
The value of $Z = 3x + 5y$ at
$O(0, 0) = 3(0) + 5(0) = 0$
$A_2(15, 0) = 3(15) + 5(0) = 45$
$P(10, 5) = 3(10) + 5(5) = 55$
$B_3(0, 5) = 3(0) + 5(5) = 25$
Hence, maximum $Z = 55$ at $x = 10$ and $y = 5$
$x + 2y = 20, x + y = 15, x = y = 0$
Region represented by $x + 2y = 20:$
Line x + 2y = 20 meets coordinate axes at $A_1(20, 0)$ and $B_1(0, 10)$, clearly, (0, 0) satisfies $\text{x}+2\text{y}\leq20$, so region containing origin represents $\text{x}+2\text{y}\leq20$ in xy -plane.
Region represented by $\text{x}+\text{y}\leq15$:
Line x + y = 15 meets coordinate axes at $A_2(15, 0)$ and $B_2(0, 15)$, clearly, (0, 0) satisfies $\text{x}+\text{y}\leq15$, so region containing origin represents x + y = 15 in xy-plane.
Region represented by $\text{y}\leq5$:
Line y = 5 is parallel to x-axis and meets at $B_3(0, 5)$ on y-axis.
Clearly (0, 0) satisfies $\text{y}\leq5$, so region containing origin represents y s 5 in xy-plane.
Region represented by $\text{x},\text{y}\geq0$:
It represent the first quadrant in xy-plane.
So, shaded region$ OA_2PB_3$ represents the feasible region.
Coordinate of P(10, 5) is obtained by solving $x + 2y = 20$ and $y - 5$
The value of $Z = 3x + 5y$ at
$O(0, 0) = 3(0) + 5(0) = 0$
$A_2(15, 0) = 3(15) + 5(0) = 45$
$P(10, 5) = 3(10) + 5(5) = 55$
$B_3(0, 5) = 3(0) + 5(5) = 25$
Hence, maximum $Z = 55$ at $x = 10$ and $y = 5$
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 the nth derivative of the following:
→$\frac{1}{x}$
Using differentials, find the approximate values of the following:
$\sin\Big(\frac{22}{14}\Big)$
→$\sin\Big(\frac{22}{14}\Big)$
Differentiate the following functions with respect to x:
$\text{x}^{\sin{\text{x}}}$
→$\text{x}^{\sin{\text{x}}}$
Solve the following differential equations:
$\text{x}\frac{\text{dy}}{\text{dx}}=\text{x + y}$
→$\text{x}\frac{\text{dy}}{\text{dx}}=\text{x + y}$
Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}49&1&6\\39&7&4\\26&2&3 \end{vmatrix}$
→$\begin{vmatrix}49&1&6\\39&7&4\\26&2&3 \end{vmatrix}$
If the lines $\frac{\text{x}-1}{-3}=\frac{\text{y}-2}{-2\text{k}}=\frac{\text{z}-3}{2}$ and $\frac{\text{x}-1}{\text{k}}=\frac{\text{y}-2}{1}=\frac{\text{z}-3}{5}$ are perpendicular, find the value of k and, hence find the equation of the plane containing these lines.
→Let A be the set of all human beings in a town at a particular time. Determine whether the following relations are reflexive, symmetric and transitive:
R = {(x, y): x and y live in the same locality}
R = {(x, y): x and y live in the same locality}
Differentiate the following functions with respect to x:
$\log(\text{cosec x}-\cot\text{x})$
→$\log(\text{cosec x}-\cot\text{x})$
Evaluate the following integrals:$\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\sin^2\text{x dx}$
→A die is tossed twice. A 'success' is getting an odd number on a toss. Find the variance of the number of successes.