Question
Maximum Z = 3x + 4y
Subject to
$2\text{x}+2\text{y}\leq80$
$2\text{x}+4\text{y}\leq120$
Subject to
$2\text{x}+2\text{y}\leq80$
$2\text{x}+4\text{y}\leq120$
✓
Answer
We have to maximize Z = 3x + 4y First, we will convert the given inequations into equations, we obtain the following equations: 2x + 2y = 80, 2x + 4y = 120 Region represented by 2x + 2y ≤ 80: The line 2x + 2y = 80 meets the coordinate axes at A(40, 0) and B(0, 40) respectively. By joining these points we obtain the line 2x + 2y = 80. Clearly (0, 0) satisfies the inequation 2x + 2y ≤ 80. So,the region containing the origin represents the solution set of the inequation 2x + 2y ≤ 80. Region represented by 2x + 4y ≤ 120: The line 2x + 4y = 120 meets the coordinate axes at C(60, 0) and D(0, 30) respectively. By joining these points we obtain the line 2x + 4y ≤ 120. Clearly (0, 0) satisfies the inequation 2x + 4y ≤ 120. So,the region containing the origin represents the solution set of the inequation 2x + 4y ≤ 120. The feasible region determined by the system of constraints, 2x + 2y ≤ 80, 2x + 4y ≤ 120 are as follows:
| Corner point | Z = 3x +4y |
| O(0, 0) | 3 × 0 + 4 × 0 = 0 |
| A(40, 0) | 3 × 40 + 4 × 0 = 120 |
| E(20, 20) | 3 × 20 + 4 × 20 = 140 |
| D(0, 30) | 10 × 0 + 4 × 30 = 120 |
We see that the maximum value of the objective function Z is 140 which is at E(20, 20) that means at x = 20 and y = 20. Thus, the optimal value of Z is 140.
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 equations of the tangent and the normal to the following curves at the indicated points.
$\text{x}^{\frac{2}{3}}+\text{y}^{\frac{2}{3}}=2\text{ at }(1,1)$
→$\text{x}^{\frac{2}{3}}+\text{y}^{\frac{2}{3}}=2\text{ at }(1,1)$
If the interest is compounded continuously at 6% per annum, how much worth Rs 100 will be after 10 years? How long will it take to double Rs 1000?
Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹ x each, ₹ y each and ₹ z each the three respectively values to its 3, 2 and 1 students with a total award money of ₹ 1,000. School Q wants to spend ₹ 1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is ₹ 600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.
find the area of the region bounded by y = |x - 1| and y = 1.
Find the area enclosed by the parabolas y= 4x - x2 and y = x2 - x.
Using integration, find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32.
Evaluvate the following intregals
$\int\frac{2\text{x}+1}{\sqrt{\text{x}^2+4\text{x}+3}}\text{dx}$
→$\int\frac{2\text{x}+1}{\sqrt{\text{x}^2+4\text{x}+3}}\text{dx}$
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=4^{\sin\text{x}}\text{ on }[0,\pi]$
→$\text{f}(\text{x})=4^{\sin\text{x}}\text{ on }[0,\pi]$
Evaluate the following integrals:
$\int_{0}^\limits{1}\frac{1}{1+2\text{x}+2\text{x}^2+2\text{x}^3+\text{x}^4}\text{ dx}$
→$\int_{0}^\limits{1}\frac{1}{1+2\text{x}+2\text{x}^2+2\text{x}^3+\text{x}^4}\text{ dx}$
Using the property of determinants and without expanding, prove that:
$\begin{vmatrix}b+c&q+r&y+z\\c+a&r+p&z+x\\a+b&p+q&x+y\end{vmatrix}=2\begin{vmatrix}a&p&x\\b&q&y\\c&r&z\end{vmatrix}$
→$\begin{vmatrix}b+c&q+r&y+z\\c+a&r+p&z+x\\a+b&p+q&x+y\end{vmatrix}=2\begin{vmatrix}a&p&x\\b&q&y\\c&r&z\end{vmatrix}$