Respuesta :
Answer:
Side length and perimeter of 1 face
Area of 1 face and surface area
Step-by-step explanation:
Suppose you are given cube with side length of x units.
Then
Side length = x units
Perimeter = 4x units
Area of 1 face [tex]=x^2[/tex] square units
Surface area [tex]=6x^2[/tex] square units
Volume [tex]=x^3[/tex] cubic units
A linear relationship is any equation that, when graphed, gives you a straight line.
Consider all options:
A. Side length and perimeter of 1 face is a linear relationship, because the graph of the function [tex]y=kx[/tex] is a straight line.
B. Perimeter of 1 face and area of 1 face is not a linear relationship, because the graph of this relationship is a quadratic parabola with equation [tex]y=kx^2[/tex].
C. Surface area and volume is not a linear relationship, because the graph of this relationship is a curve with equation [tex]y=kx^{\frac{3}{2}}[/tex].
D. Area of 1 face and surface area is a linear relationship, because the graph of the function [tex]y=kx[/tex] is a straight line.
E. Side length and volume is not a linear relationship, because the graph of this relationship is a cubic parabola with equation [tex]y=kx^3[/tex].
The pairs of variables having linear relationship are:
- side length and perimeter of 1 face f a cube
- area of 1 face and surface area of a cube
When are two quantities called as linearly related?
Suppose the first quantity's measurements are assumed by the variable 'x', and of the second quantity's measurements by the variable 'y'
Then, we say that x and y are linearly related if they can be written in the form
[tex]y = mx + c[/tex]
where m and c are constant finite numbers.
When we plot their values on Cartesian plane, they form a straight line shape, that is why called linear.
For the given cases, assuming we're taking about cube, checking if the two specified quantities are linearly related or not:
- Case 1: side length and perimeter of 1 face
Let we have:
- side length = x,
- perimeter = y
- Perimeter of one face of cube = 4x
Thus, y = 4x (thus, linearly related). (its plot is attached below)
- Case 2: perimeter of 1 face and area of 1 face
Let we have:
- side length = x
- Perimeter of 1 face = 4x = P (say)
- Area of 1 face = [tex]x^2[/tex] = A (say)
Then, we get:
[tex]A= x^2 = 4x \times x/4 = 4x \times 4x/16 = P \times \dfrac{P}{16} \\A= \dfrac{1}{16}P^2[/tex]
So there is a squared variable. Thus, the relationship between perimeter of 1 face of a cube and that face's area are not linearly related.
- Case 3: surface area and volume
Let we have:
- side length = x
- Area of cube's surface = [tex]6x^2[/tex] = A (say)
- Volume of cube = [tex]x^3 = 6x^2 \times \dfrac{1}{6\sqrt{6}} \times \sqrt{6x^2} = \dfrac{A\sqrt{A}}{6\sqrt{6}}[/tex]
Thus, the relationship between surface area of a cube and its volume are not linearly related.
- Case 4: area of 1 face and surface area
Let we have:
- side length = x
- Area of 1 face = [tex]x^2[/tex] = A (say)
- Surface area = S = area of 6 faces = [tex]6x^2 = 6 A[/tex]
Thus, we get: [tex]S= 6A[/tex]
Comparing it with y = mx + c or S = mA +c (as variables are now S and A), we see that m = 6, and c = 0.
Thus, the relationship between area of 1 face of a cube and surface area of the cube are not linearly related.
- Case 2: side length and volume
Let we have:
- side length = x
- volume of the cube = y = [tex]x^3[/tex]
Then, it cant satisfy the relation y = mx +c as there is cubic power (3 as the power) of x.
Thus, the relationship between side length and volume of a cube are not linearly related.
Thus, the pairs of variables having linear relationship are:
- side length and perimeter of 1 face f a cube
- area of 1 face and surface area of a cube
Learn more about linear relationships here:
https://brainly.com/question/16976065
