MHB Solving Inequalities and Quadratic story problems

AI Thread Summary
The discussion revolves around solving inequalities and quadratic story problems. The first question involves a dietitian needing to create a corn-bean dish with specific nutritional and cost constraints, leading to the formulation of inequalities related to the amounts of corn and beans. Key inequalities include ensuring at least 40 grams of carbs, a maximum cost of $0.36 per serving, and maintaining a ratio where the amount of corn is equal to or greater than the amount of beans. The second question pertains to calculating the speed of a train based on travel time and distance, requiring the establishment of equations for both legs of the journey. Participants are encouraged to derive the necessary inequalities and equations while seeking guidance for better understanding.
Cup0fDOOM
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Question 1:

A hospital dietitian wishes to prepare a corn-bean vegetable dish which will provide at least 40 grams of carbs, and cost no more than \$0.36 per serving.

28 grams of corn provide 8 grams of carbs and cost \$.04.

30 grams of beans provide 5 grams of carbs and cost \$.03. For taste there must be at least 56 grams of beans. There must be at least as much corn as beans.

I need to find 5 inequalities and graph them, I know how to graph and solve the inequalities I'm just having a really hard time finding the inequalities.

What I have so far: corn > or equal to 56, amount of Corn > or equal to amount of Beans.

Question 2:

Montenia takes a commuter train to her restaurant every afternoon, traveling 25 miles. Later that evening she returns home on the same train, except it is able to average 5mph faster. Montenia spend a total of 1 hour and 50 minutes total on the trains commuting. what is the speed of the train on the return trip.

I know that the 25 is the constant term but I'm having issues with the other pieces of the problem.

What I would really like is for someone to walk be through it so I can understand it myself, so in the future I won't need someone to hold my hand.
 
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Hi Cup0fDOOM,

Welcome to MHB! :)

I'll help you out with #2.

We start with $d=rt$.

a) To model the trip there we can say that $d=r_1 \times t_1$ and on the way back we can say that $d=r_2 \times t_2$. Well now we have two equations but 5 variables, so we need to make some substitutions!

b) We know that [math]t_1+t_2=\frac{11}{6}[/math], making note that 1 hour plus 5/6 of another hour is 11/6 hours.

c) We notice as well that $r_1+5=r_2$. With the previous two equations we can reduce the two times into one time and the two rates of travel into 1 rate. Lastly plug in 25 for $d$ and you should be able to solve.

Try that out and if you have any problems let me know :)

Jameson
 
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Cup0fDOOM said:
Question 1:

A hospital dietitian wishes to prepare a corn-bean vegetable dish which will provide at least 40 grams of carbs, and cost no more than \$0.36 per serving.

28 grams of corn provide 8 grams of carbs and cost \$.04.

30 grams of beans provide 5 grams of carbs and cost \$.03. For taste there must be at least 56 grams of beans. There must be at least as much corn as beans.

I need to find 5 inequalities and graph them, I know how to graph and solve the inequalities I'm just having a really hard time finding the inequalities.

What I have so far: corn > or equal to 56, amount of Corn > or equal to amount of Beans.

Introduce two variables \(b\) and \(c\) for the number of grams per serving of beans and corn respectivly.

Then starting from the bottom:

There must be at least as much corn as beans: \(c \ge b \), or rearranging \( c-b\ge 0\)

There must be at least 56 grams of beans: \( b\ge 56 \)

Now you need inequalities for cost and carbs, try to do these yourself and if you have trouble post again in this thread for more help.

Also there must be a non-negative quantity of corns, so \(c \ge 0\)

CB
 
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