Answer:
1) The answer is A) 32
2) The answer is D) 8
3) The answer is D) p(v)= 4500/v
4) The answer is A) 61.560
5) The answer is D) 0.55
Step-by-step explanation:
1) Two numbers are directly proportional when increasing one of the values increases the other proportionally. That is, if by multiplying or dividing one of them by a number, the other is also multiplied or divided by that same number. This is shown through the following relationship:
[tex]\frac{y}{x} = k[/tex]
where is a constant number and y=f(x).
In this case we have to use proportions. This is:
[tex]\frac{y1}{x1} =\frac{y2}{x2}[/tex]
That is to say, being directly proportional, the relations must result the same constant number. That's why they match as shown.
For this case, y1=f(x)=160, x1=40, x2=8 and y2 is unknown. So:
[tex]\frac{160}{40} =\frac{y2}{8}[/tex]
So [tex]4=\frac{y2}{8}[/tex].
The number 8 passes to the other side of equality with the opposite operation, that is, multiplication. So you get y2. The result is 32.
Another way of thinking the exercise is with The Rule of Three:
40 ⇒ 160
80 ⇒ x
So [tex]x=\frac{80 x 160}{40}[/tex]
2) This case is similar to the previous one. But they are not directly proportional, they are inversely proportional. A relationship must also be accomplished, but in this case the relationship is
x*y= k where is a constant number and y=f(x).
The relations between number must result the same constant number. That's why they match as shown:
x1*y1=x2*y2
For this case, y1=f(x)=2, x1=16, x2=4 and y2 is unknown. So:
16*2=4*y2. So 32=4*y2. The number 4 passes to the other side of equality with the opposite operation, that is, the division. So you get y2. The result is 8.
3)
In this case the relationship is inversely proportional. Then what is written in exercise 2 is accomplished.
To solve the exercise you must first find the constant of proportionality k. For that both numbers are multiplied. So 15*300=4500 ⇒ k=4500
Finally, it should be known that in inversely proportional functions if one of the variables increases, the other decreases; and if one of the variables decreases, the other increases. This can be represented as a function of the form: y=k/x.
For this case y= p(v), k is the calculated constant and x=v. So p(v)=4500/v
4)
You have the relationship between the sides of the shopping mall model and the actual shopping mall. But you must have a proporcion of area. The area is [tex]L^{2}[/tex]
where L is the side of the shopping mall and it's supposed to be a square.
So you have the following proporcion of area:
[tex]\frac{1}{324} =\frac{190}{x}[/tex]
where 1 and 190 are the areas of the model and 324 ([tex]18^{2}[/tex] ) and x are the areas of the actual shopping mall.
Solving the equation in a similar way to exercise 1, you get that the answer is 61.560 [tex]ft^{2}[/tex]
5)
You have a similar situation to the previous exercise. But now you must have a proporcion of volume. The volume is:
[tex]L^{3}[/tex]
where L is the side of the shopping mall and it's supposed to be a square.
So you have the following proporcion of volume:
[tex]\frac{1}{1000} =\frac{x}{550}[/tex]
where 1 and x are the areas of the model and 1000 ([tex]10^{3}[/tex] ) and 550 are the areas of the actual shopping mall.
Solving the equation, you get that the answer is 0.55 [tex]m^{3}[/tex]
