Question 1201809: Gianluca was given €50 for his birthday by his parents. He decided to buy a number of model aeroplanes at 5 euros each and a number of model cars at €2 each ,getting at least five of each . On model aeroplanes he spent €10 more than on modern cars. If he wanted to obtain as many models as possible:
A) state the 4 inequalities
b) draw the graphs of same inequalities to find how he spent his money
c) find how much change he received from 50 euros
So the 4 inequalities : y≥5 , x≥5 , 5x+2y≤50 , 5x-2y>=10
And when You plot them how do you know how he spent his money ? And how is the change worked out ?How are these questions worked out ?
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x is the number of planes.
y is the number of cars.
your constraints are:
5x + 2y <= 50
x >= 5
y >= 5
5x >= 2y + 10
your objective function is to maximize x + y.
your change is 50 minus what you spent.
using the desmos.com calculator, you are graphing the OPPOSITE of the inequalities.
the area on the graph that is not shaded is your region of feasibility.
the corner points of the feqasibility region are where your maximum or minkimum solution will lies.
the graph is shown bvelow.
the graph does not show you the results from analyzing the objective function.
that you have to do by analyzing each corner points and then picking the corner point that has the maximum value of x + y.
at (6,10), you bought 16 models.
at (5,7.5) you either bought 12 models or 13 models, because you can't buy a part of a model.
at (5,5), you bought 10 models.
at (8,5) you bought 13 models.
your maximum solution is at (6,10), where you bought 16 models.
all the constraint need to be satisfied at each corner point.
this is especially important at the corner point that has the maximum solution.
we'll just look at (6,10) to see if all the constraints are satisfied.
6*5 + 10*2 = 30 + 20 = 50, so the maximum amount of money spent is not greater than 50, meaning this constraitn is satisfied.
both x and y are greater than 5, so that constraint is satisfied.
5x >= 2y + 10 tells you that 30 is greater than or equal to 30, so that constraint is satisfied, because 5x is not smaller than 2y + 10.
your maximum solution is at (6,10).
there is no change because all of the 50 dollars are spent.
note that, at the point (5,7.5), the solution is either (5,7) or (5,8).
all the constraint need to be satisfied.
at (5,7), the constraint that 5x >= 2y + 10 is satisfied because 25 is greater than 24.
at(5,8), the constraint that 5x >= 2y + 10 is not satisfied because 25 is not greater than 26.
all the other constraint are satisfied at the point (5,8), but that goesn't matter because ALL of the constraints need to be satisfied at that point, and one of them is not satisfied.
at (5,7), all the constraints are satisfied.
5x + 2y becomes 5*5 + 2*7 = 39 which is less than 50.\
both x and y are greater than 5.
5x >= 2y + 10 becomes 25 >= 24
if you had chosen (5,7), then you would have had change, because 5x + 2y becomes 25 + 14 = 39 and you change you have been 50 minus 39 = 11.
but (5,7) was not your maximum solution, so that becomas irrelevant,
the maximum solution at (6,10) spends all your money, so there is no change.
note that, if the requirement was 5x = 2y + 10, then your solution would have had to lie on the line 5x = 2y + 10.
your maximum solution at (6,10) does lie on that line, because y * 5 = 30 and 2 * 10 + 10 = 30, so 5x = 2y + 10 is also satisfied.
if you were manually drawing the graph, you could also have graphed the opposite of the inequalities, or you could have graphed the inequalities.
either way should have led you to the same corner points.
using the desmos.com calculaltor, i graph the opposite of the inequalities because the region of feasibility shows up much clearer thqt way.
if you have any questions, let me know.
theo
Answer by ikleyn(52817)  (Show Source):
You can put this solution on YOUR website! .
Gianluca was given €50 for his birthday by his parents. He decided to buy a number of model aeroplanes
at 5 euros each and a number of model cars at €2 each, getting at least five of each.
On model aeroplanes he spent €10 more than on modern cars.
If he wanted to obtain as many models as possible:
A) state the 4 inequalities
b) draw the graphs of same inequalities to find how he spent his money
c) find how much change he received from 50 euros
So the 4 inequalities : y≥5 , x≥5 , 5x+2y≤50 , 5x-2y>=10
And when You plot them how do you know how he spent his money ?
And how is the change worked out ?How are these questions worked out ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This problem in the post is presented as if it is a Linear Programming problem.
But in reality, it is a regular fourth grade student arithmetic problem to be solved mentally in several lines.
Below I will show HOW.
Gianluca wants to spend 50 euros for buying models of two kinds: aeroplanes and cars.
He spends 10 euros more for aeroplane models than for car models.
To a 4-th grade student, it should be clear from this part that Gianluca should spend
30 euros for aeroplane models and 20 euros for car models.
Hence, the number of aeroplane models is 30/5 = 6 and the number of car models is 20/2 = 10.
ANSWER. 6 aeroplane models and 10 car models, which is 16, in total.
At this point, the core of the problem is done, and the rest is easy.
-------------------------
By the way, your inequality 5x - 2y >= 10 is INCORRECT.
The correct expression is this  5x - 2y = 10.
It is what the problem says, literally.
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