Inequality word problem!

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August 21st 2012, 06:32 PM
Calcgirl

Inequality word problem!

This problem is driving me crazy! Can someone give me a helping hand?

A store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?
August 21st 2012, 07:23 PM
richard1234

Re: Inequality word problem!

Let x be the number of A grade tea packages, and let y be the number of B grade packages. We want to maximize

$1.5x + 2y$

subject to the constraints

$\frac{x}{3} + \frac{y}{2} \le 45$

$\frac{2x}{3} + \frac{y}{2} \le 70$

This can be solved by drawing the latter two inequalities on a coordinate plane and using a "linear programming" technique.
August 21st 2012, 07:25 PM
Prove It

Re: Inequality word problem!

Quote:

Originally Posted by richard1234

Let x be the number of A grade tea packages, and let y be the number of B grade packages. We want to maximize

$1.5x + 2y$

subject to the constraints

$\frac{x}{3} + \frac{y}{2} \le 45$

$\frac{2x}{3} + \frac{y}{2} \le 70$

This can be solved by drawing the latter two inequalities on a coordinate plane and using a "linear programming" technique.

You will also need to draw the inequalities $\displaystyle \begin{align*} x \geq 0 \end{align*}$ and $\displaystyle \begin{align*} y \geq 0 \end{align*}$ .
August 21st 2012, 09:27 PM
Calcgirl

Re: Inequality word problem!

So about the 1.5x+2y part, how would i maximize that? Do i graph that on the same plane as well somehow?
And to find the maximum profit, I choose the point where the two lines intersect right?
August 21st 2012, 09:34 PM
Prove It

Re: Inequality word problem!

Quote:

Originally Posted by Calcgirl

So about the 1.5x+2y part, how would i maximize that? Do i graph that on the same plane as well somehow?
And to find the maximum profit, I choose the point where the two lines intersect right?

The maximum value will be at one of the corner points of your feasible region (that you get when you graph the inequalities). So substitute each of these points into your objective function to see which gives the maximum.