Question: `q013. Of the six proportionality equations y = k x, y = k x^2, y = k x^3, y = k / x, y = k / x^2, y = k / x^3, which one would apply to the surface area of the sculpture in the preceding?
Use the appropriate equation to answer the following:
If the sculpture originally had a exposed surface of are 4 square meters, what would be the surface area of the larger sculpture (per the conditions of the preceding problem)?
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Your solution:
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y = k / x^3
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4= k / 1/27
27*4= k / 1/27*27
k=108
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Good application of the general procedure, but you chose the wrong proportionality.
If the proportionality is y = k / x^3, then a larger value of x will result in a larger value of x^3, and hence in a smaller value of y. Since a bigger dimension x would imply a larger surface area, this proportionality doesn't apply.
You know that when you double the size of a square, it takes 4 squares of the original dimension to cover it.
So which of the proportionalities would quadruple the value of y when x is doubled?
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y = k x^2
When you double the square of 2 you get 4
4= k 2^2
so when x doubles 4 quads
...I still feel like i am doing something wrong
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First let me address your use of fractions (and order of operations) when you solved the equation
4= k / 1/3^3
for k.
In the first place you meant
4 = k / (1/3)^3.
k divided by 1/3 would not be k / 1 / 3, which by order of operations tells you to divide k by 1 then divide the result by 3.
k divided by 1/3 would be expressed as k / (1/3).
Of course in your equation you are cubing the 1/3. I didn't include the exponent above so we could focus on just the order of operations. Including the cube you would have to write this as
k / (1/3)^3.
This is then equal to k / (1/27), so your equation becomes
4 = k / (1/27).
Multiplying 27 by both sides of this equation doesn't help, because k / (1/27) * 27 means to divide k by 1/27, then multiply the result by 27. But k divided by 1/27 is k multiplied by 27/1, so
k / (1/27) * 27 = k * 27 * 27 = 729 k, not just k.
To solve the equation
4 = k / (1/27)
you would multiply both sides by 1/27, not by 27.
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See my notes directly above.
First be sure you understand the fractions and order of operations addressed in my first note.
Then I suggest you solve the problem in three different ways, as indicated by my second note.
If you can do this successfully you'll be well on your way to a solid mastery of the extremely important concept of proportionality.
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Now you can use the correct proportionality
y = k x^2.
There are many ways to set this up.
You set this up by saying that
y = 4 m^2 when x = 1/3
and find y when x = 1. This would be along the lines of your original thinking, but you have to specify both values of x. I'm not sure you saw that if x for the first figure is 1/3, then x for the second has to be 1.
You could alternatively set this up as
y = 4 m^2 when x = 1
and find y when x = 3. You would get a different value of k, but the final result for the volume would be the same.
I recommend that you set this up in both ways and verify that is doesn't matter whether x goes from 1 to 3, or from 1/3 to 1. You get the same result both ways.
Another way to solve the problem is to recall that
y = k x^2
implies that
y2 / y1 = (x2 / x1)^2.
For this question, what is the ratio x2 / x1, and what therefore is the ratio y2 / y1? What do you therefore conclude is the area of the larger figure?
This reasoning will lead to the same result as before.
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