a company manufactures two types of leaf blowers: an electric turbo model and a gas - powered tornado model…

a company manufactures two types of leaf blowers: an electric turbo model and a gas - powered tornado model. the companys production plan calls for the production of at least 820 blowers per month. it costs $78 to produce each turbo model and $117 to manufacture each tornado model, and the company has at most $79,560 per month to use for production. find the number of units that should be produced to maximize profit for the company, and the maximum profit, if the profit on each turbo model is $35 and the profit on each tornado model is $45.\nthe maximum value is $32700 by producing 420 units of the turbo model and 400 units of the tornado model. (type integers or decimals.)

a company manufactures two types of leaf blowers: an electric turbo model and a gas - powered tornado model. the companys production plan calls for the production of at least 820 blowers per month. it costs $78 to produce each turbo model and $117 to manufacture each tornado model, and the company has at most $79,560 per month to use for production. find the number of units that should be produced to maximize profit for the company, and the maximum profit, if the profit on each turbo model is $35 and the profit on each tornado model is $45.\nthe maximum value is $32700 by producing 420 units of the turbo model and 400 units of the tornado model. (type integers or decimals.)

Answer

Explanation:

Step1: Define variables

Let $x$ be the number of Turbo models and $y$ be the number of Tornado models.

Step2: Set up constraints

Production - quantity constraint: $x + y\geq820$. Cost - constraint: $78x + 117y\leq79560$, or $2x + 3y\leq2040$. Also, $x\geq0,y\geq0$.

Step3: Set up profit function

The profit function $P=35x + 45y$.

Step4: Rewrite inequalities for graphing

$y\geq - x+820$, $y\leq-\frac{2}{3}x + 680$.

Step5: Find intersection points

Intersection of $y=-x + 820$ and $y=-\frac{2}{3}x+680$: $-x + 820=-\frac{2}{3}x+680$. $-\ x+\frac{2}{3}x=680 - 820$. $-\frac{1}{3}x=-140$, so $x = 420$. Then $y=-420 + 820=400$. Intersection of $y=-x + 820$ and $x = 0$ gives $(0,820)$. Intersection of $y=-\frac{2}{3}x+680$ and $y = 0$ gives $(1020,0)$.

Step6: Evaluate profit function at corner - points

At $(0,820)$: $P=35\times0+45\times820=45\times820 = 36900$. At $(1020,0)$: $P=35\times1020+45\times0=35700$. At $(420,400)$: $P=35\times420+45\times400=14700 + 18000=32700$.

Answer:

The maximum value is $$32700$ by producing $420$ units of the Turbo model and $400$ units of the Tornado model.