I have a linear programming problem, but I dont think I got the constrainst right?

Question

A vase requires 25 oz of clay and 5 oz of glaze. A bowl requires 20 oz of clay and 10 oz of glaze. There are 500 oz of clay and 160 oz of glaze available. The profit on one vase is $5 and the profit of one bowl is $3. How many vases and bowls should they make to maximize profit?
I used x for the vases and y for the bowls. I think the profit equation is P=5x + 3y. and as constraints i have 25x+20y is less than or equal to 500. 5x+10y is less than or equal to 160. and i was given that x and y are greater or equal to 0. I believe there is something wrong with my first constraint.

i think something is wrong because the graph is supposed to form a triangle, and my 2 lines dont meet when i graph them

can somebody explain to me how the second answer was done? where did the 3x's come from?

Answer 1

if no of vases = x and no of bowls = y

then total clay required = 25x + 20y
total glaze required = 5x + 10y

total profit = 5x + 3y

25x + 20y <= 500
5x+10y <= 160

so, we solve 25x + 20y = 500
5x + 10y = 160
or, 5x + 4y = 100
and, x + 2y = 32

or 3x = 100- 64 = 36
or, x = 12

and y = (32-12)/2 = 10

so one vertice is (12,10)

for x = 0, from eq-i, y = 25 , from eq-ii, y= 16
we get another vertice (0, 16)

for y = 0 from eq-i, x = 20, from eq-ii, x = 32
we get another vertice (20, 0)

now we calculate profit P,
(20,0) -> P = 100
(0, 16) -> P = 48
(12,10) -> P = 60 + 30 = 90

So profit will be maximized if only vase is produced. What's the problem?

Answer 2

Theres nothing wrong with the constraints you are most likely graphing the question wrong. Try redrawing it. I did it and the answer is that you should be making 20 vases. the above answer is also correct but it takes more time and work. graphing is much easier and takes less time.
Its up to you.
whatever you feel is easier.

Answer 3

I don't see anything wrong with it. What makes you think it's wrong?