4. **Stating the problem:** Find the maximum and minimum values of \(P=3x+2y\) subject to the inequalities \(x \geq 2.5\), \(y \geq 3.5\), and \(y \leq -4x + 11.5\). 1. Identify vertices of feasible region formed by the inequalities: - Intersection of \(x=2.5\) and \(y=3.5\): \((2.5,3.5)\) - Intersection of \(x=2.5\) and \(y=-4x+11.5\): $$y=-4(2.5)+11.5=-10+11.5=1.5$$ But \(y \geq 3.5\), so \(y=1.5\) is outside feasible region, discard. - Intersection of \(y=3.5\) and \(y=-4x+11.5\): $$3.5=-4x+11.5 \Rightarrow 4x = 11.5-3.5=8 \Rightarrow x=2$$ But \(x \geq 2.5\), so this point is outside feasible region. Thus vertices are: - \(A=(2.5,3.5)\) - Check boundary beyond \(x=2.5\) for intersection of \(y \geq 3.5\) and \(y \leq -4x+11.5\). 2. Check endpoint of \(x\) range and feasible \(y\): At \(x=2.5\), \(y=3.5\) is feasible. When \(x=2.5\), \(y=-4(2.5)+11.5=1.5\) which is less than\(3.5\), so upwards boundary is \(y=3.5\). 3. Check when \(y=3.5\) meets \(x \geq 2.5\) and may vary. Since \(y\) is bounded above by \(y= -4x + 11.5\) which decreases with \(x\), the feasible region is between \(y=3.5\) and \(y = -4x + 11.5\) starting at \(x=2.5\). 4. Find intersections: At \(x=2.5\), feasible point is \((2.5,3.5)\). As \(x\) increases, \(y\) can go down to \(y= -4x + 11.5\). Check when \(y=3.5\) equals \(y=-4x + 11.5\):\(x=2\) outside feasible region. So boundary points for maximizing and minimizing P should be among vertices: - Point A: \((2.5,3.5)\) - Point B: The intersection of \(y=-4x + 11.5\) and \(x=2.5\) but rejected due to y < 3.5 Thus the feasible region is a polygon bounded by points: - \(A=(2.5,3.5)\) - \(C=(2.5,\text{some limit})\), but limited by given inequalities Since no upper x-bound is given, maximize and minimize by analyzing at vertex \(A\) and where \(P\) is minimized on feasible boundary. Calculate \(P\) at \(A\): $$P=3(2.5)+2(3.5)=7.5+7=14.5$$ At higher \(x\), \(P=3x+2y \leq 3x + 2(-4x+11.5) = 3x - 8x + 23 = -5x + 23\) When \(x\) is minimum 2.5, \(P = -5(2.5)+23=10.5\), when \(x\) increases, \(P\) decreases. Thus, - Maximum \(P=14.5\) at \(x=2.5,y=3.5\) - Minimum \(P\) tends to decrease beyond x=2.5; no maximum x given so minimum tends to \(-\infty\) Since no upper bound on \(x\) is given, minimum value of \(P\) is unbounded below. --- 5. **Problem:** Optimize \(P=x+y\) with \(x \geq 5\), \(y \geq 8\), and \(y \leq 3x + 5\). Vertices of feasible region: - Intersection of \(x=5\) and \(y=8\): \( (5,8) \) - Intersection of \(x=5\) and \(y=3(5)+5=20\): \( (5,20) \) - Intersection of \(y=8\) and \(y=3x+5\): $$8 = 3x + 5 \Rightarrow 3x = 3 \Rightarrow x=1$$ But \(x \geq 5\), discard this point. Vertices: - \(A=(5,8)\) - \(B=(5,20)\) Evaluate \(P\) at \(A\): \(5+8=13\) Evaluate \(P\) at \(B\): \(5+20=25\) No upper bound on \(x\), but since \(y \leq 3x+5\), for larger \(x\), \(P = x + y \leq x + 3x + 5 = 4x + 5\) which grows as \(x\) increases. Thus, no maximum finite value; minimum at \(13\). --- 6. **Problem:** Find max and min of \(P=2(x+y)\) with \(x \geq 2.5\), \(y \geq 10.5\), and \(y \leq 2x + 7.5\). Vertices: - Intersection of \(x=2.5\) and \(y=10.5\): \((2.5,10.5)\) - Intersection of \(x=2.5\) and \(y=2(2.5)+7.5=5+7.5=12.5\): \((2.5,12.5)\) - Intersection of \(y=10.5\) and \(y=2x+7.5\): $$10.5=2x+7.5 \Rightarrow 2x=3 \Rightarrow x=1.5$$ But \(x \geq 2.5\) discard. Vertices: - \(A = (2.5,10.5)\) - \(B = (2.5,12.5)\) Evaluate \(P\) at \(A\): $$P=2(2.5+10.5)=2(13)=26$$ At \(B\): $$P=2(2.5+12.5)=2(15)=30$$ No upper bound on \(x\), but \(y \leq 2x +7.5\) grows linearly. As \(x\) increases, max \(P\) tends to infinity, minimum at \(26\). --- 7. **Music vendor problem:** (a) Inequalities: - Total cost: \(90x + 45y \leq 855\) - CDs: more than one but no more than seven: $$1 < x \leq 7$$ or \(x \, > 1\) and \(x \leq 7\) - Cassettes: at least three $$y \geq 3$$ (b) Graph: Axes with units equal to variables; draw lines for \(90x + 45y = 855\), \(x=1\), \(x=7\), \(y=3\), shade feasible region. (c) Profit: (i) Total profit: $$P = 20x + 8y$$ (ii) Evaluate \(P\) at vertices of feasible region: Vertices from intersection of lines: - Intersection of \(x=1\) and \(y=3\): feasible? $$90(1) + 45(3)=90 + 135=225 \leq 855$$ yes. - Intersection of \(x=7\) and \(y=3\): $$90(7) + 45(3)=630 + 135=765 \leq 855$$ yes. - Intersection of budget line \(90x + 45y=855\) and \(x=1\): $$90(1)+45y=855 \Rightarrow 45y=765 \Rightarrow y=17$$ feasible as \(y \, \geq 3\). - Intersection of budget line and \(x=7\): $$90(7) + 45y = 855 \Rightarrow 630 + 45y=855 \Rightarrow 45y=225 \Rightarrow y=5$$ Profit at: - (1,3): \(20(1)+8(3)=20+24=44\) - (7,3): \(20(7)+8(3)=140+24=164\) - (1,17): \(20(1)+8(17)=20+136=156\) - (7,5): \(20(7)+8(5)=140+40=180\) Maximum profit is \(180\) at \((7,5)\). --- 8. **Student buying jerseys and shirts:** (a) Inequalities: - Budget: \(25x + 35y \leq 265\) - Jerseys: at least 2 but no more than 5 $$2 \leq x \leq 5$$ - Shirts: more than 1 $$y > 1$$ (b) Graph: plot lines \(25x+35y=265\), vertical lines \(x=2\), \(x=5\), and \(y=1\). (c) Profit: (i) \(P = 5x + 8y\) (ii) Find vertices: - Intersection \(x=2, y=1\): check budget: $$25(2) + 35(1)=50+35=85 \leq 265$$ yes - Intersection \(x=5, y=1\): $$25(5)+35(1)=125+35=160\) yes - Intersection of budget line \(25x+35y=265\) and \(x=2\): $$25(2)+35y=265 \Rightarrow 50+35y=265 \Rightarrow 35y=215 \Rightarrow y=6.14$$ - Intersection of budget line and \(x=5\): $$25(5)+35y=265 \Rightarrow 125+35y=265 \Rightarrow 35y=140 \Rightarrow y=4$$ Profit at: - (2,1): \(5(2)+8(1)=10+8=18\) - (5,1): \(25+8=33\) - (2,6.14): \(10+8(6.14)=10+49.12=59.12\) - (5,4): \(25+32=57\) Maximum profit \(\approx 59.12\) at \( (2,6.14) \) --- 9. **Butcher buying cows and pigs:** (a) Inequalities: - Budget: \(900x + 700y \leq 9800\) - Pigs fewer than cows: $$y < x$$ - Space limits total: $$x + y \leq 12$$ (b) Graph: plot budget line, space line, and \(y=x\). (c) Profit: $$P = 550x + 325y$$ Vertices: - Intersection of space and budget: Solve: $$x + y = 12$$ $$900x + 700y = 9800$$ From space: $$y = 12 - x$$ Substitute: $$900x + 700(12 - x) = 9800$$ $$900x + 8400 - 700x = 9800$$ $$200x = 1400 \Rightarrow x=7$$ Then \(y=5\) Check \(y < x\): \(5 < 7\) valid. - Intersect budget with \(y = x\): $$900x + 700x = 9800 \Rightarrow 1600x = 9800 \Rightarrow x=6.125$$ \(y=6.125\) Check space: \(x+y=12.25 > 12\) invalid. Evaluate profit at \( (7,5) \): $$P = 550(7) + 325(5) =3850 + 1625=5475$$ Other corners: - \(x=0, y=0\) profit = 0 - \(x=0, y=12\) not valid since \(y 2x$$ (b) Graph these inequalities. (c) Income: $$I = 2500x + 4500y$$ Vertices: - Using \(x = 10\), \(y > 20\) and \(x + y \leq 35\): possible \(y\) ranges 21 to 25. Check profits at: - \((10, 21)\): \(2500(10)+4500(21)=25000 + 94500=119500\) - \((10,25)\): \(25000 + 112500=137500\) - Intersect \(y=2x\) and \(x+y=35\): $$y=2x\Rightarrow x+2x=35 \Rightarrow 3x=35 \Rightarrow x=11.67, y=23.33$$ Profit: $$2500(11.67)+4500(23.33)=29175+104985=134160$$ Maximum income is at \( (10, 25) \) : \(137500\).