What does a proportion look like?
1 answer:
A proportion looks like this:
It means that the quantity a,b,c and d are in the same proportion: the ratio between a and b is the same ratio between c and d.
They are particularly useful in situation when you have an example given, and you want to extrapolate the relationship for another couple.
For example, you can think of the following problem: "Five apples cost 2 dollars. How much will 8 apples cost?"
You can build the proportion
And then solve it for x:
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Answer: of quinoa,
of flaxseed,
of soy protein.
Step-by-step explanation:
<h3> The missing table is attached.</h3>Let be "q" the amount of cups of quinoa you'd need to use making the granolas with 3 cups of oats.
Based on the table you can write the following proportion:
Solving for "q", you get:
Let be "f" the amount of cups of flaxseed you'd need to use making the granolas with 3 cups of oats.
With the data given in the table you can write the following proportion:
Solving for "f", you get:
Let be "s" the amount of cups of soy protein you'd need to use making the granolas with 3 cups of oats.
Using the table you can write this proportion:
Solving for "s", you get:
For given f(x, y) the extremum: (12, 24) which is the minimum.
For given question,
We have been given a function f(x) = 4x² + 2y² under the constraint 3x+3y= 108
We use the constraint to build the constraint function,
g(x, y) = 3x + 3y
We then take all the partial derivatives which will be needed for the Lagrange multiplier equations:
Setting up the Lagrange multiplier equations:
⇒ 8x = 3λ .....................(1)
⇒ 4y = 3λ ......................(2)
constraint: 3x + 3y = 108 .......................(3)
Taking (1) / (2), (assuming λ ≠ 0)
⇒ 8x/4y = 1
⇒ 2x = y
Substitute this value of y in equation (3),
⇒ 3x + 3y = 108
⇒ 3x + 3(2x) = 108
⇒ 3x + 6x = 108
⇒ 9x = 108
⇒ x = 12
⇒ y = 2 × 12
⇒ y = 24
So, the saddle point (critical point) is (12, 24)
Now we find the value of f(12, 24)
⇒ f(12, 24) = 4(12)² + 2(24)²
⇒ f(12, 24) = 576 + 1152
⇒ f(12, 24) = 1728 ................(1)
Consider point (18,18)
At this point the value of function f(x, y) is,
⇒ f(18, 18) = 4(18)² + 2(18)²
⇒ f(18, 18) = 1296 + 648
⇒ f(18, 18) = 1944 ..............(2)
From (1) and (2),
1728 < 1944
This means, given extremum (12, 24) is minimum.
Therefore, for given f(x, y) the extremum: (12, 24) which is the minimum.
Learn more about the extremum here:
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