Math / Geometry

QuestionRectangles F and G are similar. If the area of rectangle F is 63 , what is the area of rectangle G ?
Answer Attempt 1 out of 2 $\square$ Submit Answer

Studdy Solution

STEP 1

What is this asking? We're given two similar rectangles, one with a known area, and we need to find the area of the other, knowing their lengths. Watch out! Don't just multiply or divide the areas by the ratio of the lengths!
Areas scale with the square of the ratio of corresponding sides.

STEP 2

1. Find the scale factor.
2. Find the width of rectangle F.
3. Find the width of rectangle G.
4. Calculate the area of rectangle G.

STEP 3

We know the rectangles are similar, which means their corresponding sides are proportional.
Rectangle F has a length of $9$ and rectangle G has a length of $6.3$ .
Let's call the scale factor $k$ .

STEP 4

We can set up the proportion: $9 \cdot k = 6.3$ .
We're doing this because multiplying the length of rectangle F by the scale factor should give us the length of rectangle G.

STEP 5

To find $k$ , we divide both sides of the equation by $9$ : $k = \frac{6.3}{9}$ .
This gives us $k = \textbf{0.7}$ .
So, the scale factor is $\textbf{0.7}$ .
This means the sides of rectangle G are $0.7$ times the size of the corresponding sides of rectangle F.

STEP 6

We know the area of rectangle F is $63$ , and its length is $9$ .
Let's call the width of rectangle F, $w_F$ .
The area of a rectangle is length times width, so $9 \cdot w_F = 63$ .

STEP 7

To find $w_F$ , we divide both sides by $9$ : $w_F = \frac{63}{9}$ .
This gives us $w_F = \textbf{7}$ .
So, the width of rectangle F is $\textbf{7}$ .

STEP 8

Since the rectangles are similar, the width of rectangle G, $w_G$ , is the width of rectangle F multiplied by the scale factor: $w_G = w_F \cdot k$ .

STEP 9

We know $w_F = 7$ and $k = 0.7$ , so $w_G = 7 \cdot 0.7 = \textbf{4.9}$ .
The width of rectangle G is $\textbf{4.9}$ .

STEP 10

Now we have the length and width of rectangle G: $6.3$ and $4.9$ .
The area is length times width, so the area of rectangle G is $6.3 \cdot 4.9$ .

STEP 11

Multiplying these together, we get $6.3 \cdot 4.9 = \textbf{30.87}$ .

STEP 12

The area of rectangle G is $\textbf{30.87}$ .

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QuestionRectangles F and G are similar. If the area of rectangle F is 63 , what is the area of rectangle G ?
Answer Attempt 1 out of 2 $\square$ Submit Answer

Studdy Solution

STEP 1

What is this asking? We're given two similar rectangles, one with a known area, and we need to find the area of the other, knowing their lengths. Watch out! Don't just multiply or divide the areas by the ratio of the lengths!
Areas scale with the square of the ratio of corresponding sides.

STEP 2

1. Find the scale factor.
2. Find the width of rectangle F.
3. Find the width of rectangle G.
4. Calculate the area of rectangle G.

STEP 3

We know the rectangles are similar, which means their corresponding sides are proportional.
Rectangle F has a length of $9$ and rectangle G has a length of $6.3$ .
Let's call the scale factor $k$ .

STEP 4

We can set up the proportion: $9 \cdot k = 6.3$ .
We're doing this because multiplying the length of rectangle F by the scale factor should give us the length of rectangle G.

STEP 5

To find $k$ , we divide both sides of the equation by $9$ : $k = \frac{6.3}{9}$ .
This gives us $k = \textbf{0.7}$ .
So, the scale factor is $\textbf{0.7}$ .
This means the sides of rectangle G are $0.7$ times the size of the corresponding sides of rectangle F.

STEP 6

We know the area of rectangle F is $63$ , and its length is $9$ .
Let's call the width of rectangle F, $w_F$ .
The area of a rectangle is length times width, so $9 \cdot w_F = 63$ .

STEP 7

To find $w_F$ , we divide both sides by $9$ : $w_F = \frac{63}{9}$ .
This gives us $w_F = \textbf{7}$ .
So, the width of rectangle F is $\textbf{7}$ .

STEP 8

Since the rectangles are similar, the width of rectangle G, $w_G$ , is the width of rectangle F multiplied by the scale factor: $w_G = w_F \cdot k$ .

STEP 9

We know $w_F = 7$ and $k = 0.7$ , so $w_G = 7 \cdot 0.7 = \textbf{4.9}$ .
The width of rectangle G is $\textbf{4.9}$ .

STEP 10

Now we have the length and width of rectangle G: $6.3$ and $4.9$ .
The area is length times width, so the area of rectangle G is $6.3 \cdot 4.9$ .

STEP 11

Multiplying these together, we get $6.3 \cdot 4.9 = \textbf{30.87}$ .

STEP 12

The area of rectangle G is $\textbf{30.87}$ .

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QuestionRectangles F and G are similar. If the area of rectangle F is 63 , what is the area of rectangle G ?Answer Attempt 1 out of 2 □\square□ Submit Answer

Studdy Solution

STEP 1

What is this asking? We're given two similar rectangles, one with a known area, and we need to find the area of the other, knowing their lengths. Watch out! Don't just multiply or divide the areas by the ratio of the lengths!Areas scale with the *square* of the ratio of corresponding sides.

STEP 2

1. Find the scale factor.2. Find the width of rectangle F.3. Find the width of rectangle G.4. Calculate the area of rectangle G.

STEP 3

We know the rectangles are similar, which means their corresponding sides are proportional.Rectangle F has a length of 999 and rectangle G has a length of 6.36.36.3.Let's call the **scale factor** kkk.

STEP 4

We can set up the proportion: 9⋅k=6.39 \cdot k = 6.39⋅k=6.3.We're doing this because multiplying the length of rectangle F by the scale factor should give us the length of rectangle G.

STEP 5

To find kkk, we **divide both sides** of the equation by 999: k=6.39k = \frac{6.3}{9}k=96.3​.This gives us k=0.7k = \textbf{0.7}k=0.7.So, the **scale factor** is 0.7\textbf{0.7}0.7.This means the sides of rectangle G are 0.70.70.7 times the size of the corresponding sides of rectangle F.

STEP 6

We know the area of rectangle F is 636363, and its length is 999.Let's call the width of rectangle F, wFw_FwF​.The area of a rectangle is length times width, so 9⋅wF=639 \cdot w_F = 639⋅wF​=63.

STEP 7

To find wFw_FwF​, we **divide both sides** by 999: wF=639w_F = \frac{63}{9}wF​=963​.This gives us wF=7w_F = \textbf{7}wF​=7.So, the width of rectangle F is 7\textbf{7}7.

STEP 8

Since the rectangles are similar, the width of rectangle G, wGw_GwG​, is the width of rectangle F multiplied by the **scale factor**: wG=wF⋅kw_G = w_F \cdot kwG​=wF​⋅k.

STEP 9

We know wF=7w_F = 7wF​=7 and k=0.7k = 0.7k=0.7, so wG=7⋅0.7=4.9w_G = 7 \cdot 0.7 = \textbf{4.9}wG​=7⋅0.7=4.9.The width of rectangle G is 4.9\textbf{4.9}4.9.

STEP 10

Now we have the length and width of rectangle G: 6.36.36.3 and 4.94.94.9.The area is length times width, so the area of rectangle G is 6.3⋅4.96.3 \cdot 4.96.3⋅4.9.

STEP 11

Multiplying these together, we get 6.3⋅4.9=30.876.3 \cdot 4.9 = \textbf{30.87}6.3⋅4.9=30.87.

STEP 12

The area of rectangle G is 30.87\textbf{30.87}30.87.

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QuestionRectangles F and G are similar. If the area of rectangle F is 63 , what is the area of rectangle G ?
Answer Attempt 1 out of 2 $\square$ Submit Answer

What is this asking? We're given two similar rectangles, one with a known area, and we need to find the area of the other, knowing their lengths. Watch out! Don't just multiply or divide the areas by the ratio of the lengths!
Areas scale with the square of the ratio of corresponding sides.

1. Find the scale factor.
2. Find the width of rectangle F.
3. Find the width of rectangle G.
4. Calculate the area of rectangle G.

We know the rectangles are similar, which means their corresponding sides are proportional.
Rectangle F has a length of $9$ and rectangle G has a length of $6.3$ .
Let's call the scale factor $k$ .

We can set up the proportion: $9 \cdot k = 6.3$ .
We're doing this because multiplying the length of rectangle F by the scale factor should give us the length of rectangle G.

To find $k$ , we divide both sides of the equation by $9$ : $k = \frac{6.3}{9}$ .
This gives us $k = \textbf{0.7}$ .
So, the scale factor is $\textbf{0.7}$ .
This means the sides of rectangle G are $0.7$ times the size of the corresponding sides of rectangle F.

We know the area of rectangle F is $63$ , and its length is $9$ .
Let's call the width of rectangle F, $w_F$ .
The area of a rectangle is length times width, so $9 \cdot w_F = 63$ .

To find $w_F$ , we divide both sides by $9$ : $w_F = \frac{63}{9}$ .
This gives us $w_F = \textbf{7}$ .
So, the width of rectangle F is $\textbf{7}$ .

Since the rectangles are similar, the width of rectangle G, $w_G$ , is the width of rectangle F multiplied by the scale factor: $w_G = w_F \cdot k$ .

We know $w_F = 7$ and $k = 0.7$ , so $w_G = 7 \cdot 0.7 = \textbf{4.9}$ .
The width of rectangle G is $\textbf{4.9}$ .

Now we have the length and width of rectangle G: $6.3$ and $4.9$ .
The area is length times width, so the area of rectangle G is $6.3 \cdot 4.9$ .

Multiplying these together, we get $6.3 \cdot 4.9 = \textbf{30.87}$ .

The area of rectangle G is $\textbf{30.87}$ .