Solved on Feb 01, 2024

Fill in the table for side length and area of different squares. Is the relationship between side length and area proportional?
$3$ cm, $\qquad \qquad \qquad \qquad \qquad$ $100$ cm, $\qquad \qquad \qquad \qquad \qquad$ $25$ cm, $\qquad \qquad \qquad \qquad \qquad$ $s$ cm, $\qquad \qquad \qquad \qquad \qquad$

STEP 1

Assumptions
1. The relationship between the side length and the area of a square is given by the formula $Area = (side\ length)^2$ .
2. We are given the side lengths of different squares and need to calculate their respective areas.
3. We are also asked to determine if the relationship between side length and area is proportional.

STEP 2

To fill in the table for the first square with a side length of 3 cm, apply the formula for the area of a square.
$Area = (side\ length)^2$

STEP 3

Calculate the area for a side length of 3 cm.
$Area = (3\ cm)^2$

STEP 4

Perform the calculation.
$Area = 3\ cm \times 3\ cm = 9\ cm^2$

STEP 5

To fill in the table for the second square with a side length of 100 cm, apply the formula for the area of a square.
$Area = (100\ cm)^2$

STEP 6

Perform the calculation.
$Area = 100\ cm \times 100\ cm = 10,000\ cm^2$

STEP 7

To fill in the table for the third square with a side length of 25 cm, apply the formula for the area of a square.
$Area = (25\ cm)^2$

STEP 8

Perform the calculation.
$Area = 25\ cm \times 25\ cm = 625\ cm^2$

STEP 9

To express the area of a square with an unknown side length $s$ , apply the formula for the area of a square.
$Area = s^2$

STEP 10

Now that we have calculated the areas for the given side lengths and expressed the area for an unknown side length $s$ , we can fill in the table.
\begin{tabular}{|c|c|} \hline side length $(\mathrm{cm})$ & area $\left(\mathrm{cm}^{2}\right)$ \\ \hline 3 & 9 \\ \hline 100 & 10,000 \\ \hline 25 & 625 \\ \hline $s$ & $s^2$ \\ \hline \end{tabular}

STEP 11

To answer part b of the question, we need to understand what a proportional relationship is. A proportional relationship between two quantities is one where the ratio of one quantity to the other is constant.

STEP 12

Check if the ratio of the area to the side length squared is constant for the given side lengths.

STEP 13

Calculate the ratio for the first square with a side length of 3 cm.
$Ratio = \frac{Area}{(side\ length)^2} = \frac{9\ cm^2}{(3\ cm)^2} = \frac{9\ cm^2}{9\ cm^2} = 1$

STEP 14

Calculate the ratio for the second square with a side length of 100 cm.
$Ratio = \frac{Area}{(side\ length)^2} = \frac{10,000\ cm^2}{(100\ cm)^2} = \frac{10,000\ cm^2}{10,000\ cm^2} = 1$

STEP 15

Calculate the ratio for the third square with a side length of 25 cm.
$Ratio = \frac{Area}{(side\ length)^2} = \frac{625\ cm^2}{(25\ cm)^2} = \frac{625\ cm^2}{625\ cm^2} = 1$

STEP 16

Since the ratio of the area to the side length squared is constant (equal to 1) for all given side lengths, we can conclude that the relationship between the side length of a square and the area of a square is proportional.
The filled table is as follows:
\begin{tabular}{|c|c|} \hline side length $(\mathrm{cm})$ & area $\left(\mathrm{cm}^{2}\right)$ \\ \hline 3 & 9 \\ \hline 100 & 10,000 \\ \hline 25 & 625 \\ \hline $s$ & $s^2$ \\ \hline \end{tabular}
And the relationship between the side length of a square and the area of a square is proportional.

Was this helpful?

STEP 1

STEP 2

To fill in the table for the first square with a side length of 3 cm, apply the formula for the area of a square.
$Area = (side\ length)^2$

STEP 3

Calculate the area for a side length of 3 cm.
$Area = (3\ cm)^2$

STEP 4

Perform the calculation.
$Area = 3\ cm \times 3\ cm = 9\ cm^2$

STEP 5

To fill in the table for the second square with a side length of 100 cm, apply the formula for the area of a square.
$Area = (100\ cm)^2$

STEP 6

Perform the calculation.
$Area = 100\ cm \times 100\ cm = 10,000\ cm^2$

STEP 7

To fill in the table for the third square with a side length of 25 cm, apply the formula for the area of a square.
$Area = (25\ cm)^2$

STEP 8

Perform the calculation.
$Area = 25\ cm \times 25\ cm = 625\ cm^2$

STEP 9

To express the area of a square with an unknown side length $s$ , apply the formula for the area of a square.
$Area = s^2$

STEP 10

STEP 11

To answer part b of the question, we need to understand what a proportional relationship is. A proportional relationship between two quantities is one where the ratio of one quantity to the other is constant.

STEP 12

Check if the ratio of the area to the side length squared is constant for the given side lengths.

STEP 13

Calculate the ratio for the first square with a side length of 3 cm.
$Ratio = \frac{Area}{(side\ length)^2} = \frac{9\ cm^2}{(3\ cm)^2} = \frac{9\ cm^2}{9\ cm^2} = 1$

STEP 14

Calculate the ratio for the second square with a side length of 100 cm.
$Ratio = \frac{Area}{(side\ length)^2} = \frac{10,000\ cm^2}{(100\ cm)^2} = \frac{10,000\ cm^2}{10,000\ cm^2} = 1$

STEP 15

Calculate the ratio for the third square with a side length of 25 cm.
$Ratio = \frac{Area}{(side\ length)^2} = \frac{625\ cm^2}{(25\ cm)^2} = \frac{625\ cm^2}{625\ cm^2} = 1$

STEP 16

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

STEP 1

STEP 2

To fill in the table for the first square with a side length of 3 cm, apply the formula for the area of a square.Area=(side length)2Area = (side\ length)^2Area=(side length)2

STEP 3

Calculate the area for a side length of 3 cm.Area=(3 cm)2Area = (3\ cm)^2Area=(3 cm)2

STEP 4

Perform the calculation.Area=3 cm×3 cm=9 cm2Area = 3\ cm \times 3\ cm = 9\ cm^2Area=3 cm×3 cm=9 cm2

STEP 5

To fill in the table for the second square with a side length of 100 cm, apply the formula for the area of a square.Area=(100 cm)2Area = (100\ cm)^2Area=(100 cm)2

STEP 6

Perform the calculation.Area=100 cm×100 cm=10,000 cm2Area = 100\ cm \times 100\ cm = 10,000\ cm^2Area=100 cm×100 cm=10,000 cm2

STEP 7

To fill in the table for the third square with a side length of 25 cm, apply the formula for the area of a square.Area=(25 cm)2Area = (25\ cm)^2Area=(25 cm)2

STEP 8

Perform the calculation.Area=25 cm×25 cm=625 cm2Area = 25\ cm \times 25\ cm = 625\ cm^2Area=25 cm×25 cm=625 cm2

STEP 9

To express the area of a square with an unknown side length sss, apply the formula for the area of a square.Area=s2Area = s^2Area=s2

STEP 10

STEP 11

To answer part b of the question, we need to understand what a proportional relationship is. A proportional relationship between two quantities is one where the ratio of one quantity to the other is constant.

STEP 12

Check if the ratio of the area to the side length squared is constant for the given side lengths.

STEP 13

Calculate the ratio for the first square with a side length of 3 cm.Ratio=Area(side length)2=9 cm2(3 cm)2=9 cm29 cm2=1Ratio = \frac{Area}{(side\ length)^2} = \frac{9\ cm^2}{(3\ cm)^2} = \frac{9\ cm^2}{9\ cm^2} = 1Ratio=(side length)2Area​=(3 cm)29 cm2​=9 cm29 cm2​=1

STEP 14

STEP 15

Calculate the ratio for the third square with a side length of 25 cm.Ratio=Area(side length)2=625 cm2(25 cm)2=625 cm2625 cm2=1Ratio = \frac{Area}{(side\ length)^2} = \frac{625\ cm^2}{(25\ cm)^2} = \frac{625\ cm^2}{625\ cm^2} = 1Ratio=(side length)2Area​=(25 cm)2625 cm2​=625 cm2625 cm2​=1

STEP 16

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

To fill in the table for the first square with a side length of 3 cm, apply the formula for the area of a square.
$Area = (side\ length)^2$

Calculate the area for a side length of 3 cm.
$Area = (3\ cm)^2$

Perform the calculation.
$Area = 3\ cm \times 3\ cm = 9\ cm^2$

To fill in the table for the second square with a side length of 100 cm, apply the formula for the area of a square.
$Area = (100\ cm)^2$

Perform the calculation.
$Area = 100\ cm \times 100\ cm = 10,000\ cm^2$

To fill in the table for the third square with a side length of 25 cm, apply the formula for the area of a square.
$Area = (25\ cm)^2$

Perform the calculation.
$Area = 25\ cm \times 25\ cm = 625\ cm^2$

To express the area of a square with an unknown side length $s$ , apply the formula for the area of a square.
$Area = s^2$

Calculate the ratio for the first square with a side length of 3 cm.
$Ratio = \frac{Area}{(side\ length)^2} = \frac{9\ cm^2}{(3\ cm)^2} = \frac{9\ cm^2}{9\ cm^2} = 1$

Calculate the ratio for the third square with a side length of 25 cm.
$Ratio = \frac{Area}{(side\ length)^2} = \frac{625\ cm^2}{(25\ cm)^2} = \frac{625\ cm^2}{625\ cm^2} = 1$