Math / Geometry

QuestionTIME RENAINING 54:50
Bobby knows that the perimeter of the original rectangle is 120 meters. He also knows that the perimeter of the reduced rectangle is 30 meters and the reduced length is 9 meters.
Not drawn to scale What is the width of the original rectangle? 20 meters 24 meters 36 meters 48 meters Mark this and return Save and Exit

Studdy Solution

STEP 1

What is this asking? We're looking for the original rectangle's width, knowing its perimeter and the smaller rectangle's perimeter and length. Watch out! Don't mix up the original and reduced rectangle dimensions!

STEP 2

1. Find the reduced width
2. Find the reduced scale
3. Find the original length
4. Calculate the original width

STEP 3

We know the reduced rectangle's perimeter is $30$ meters and its length is $9$ meters.
Let's use the perimeter formula: $P = 2 \cdot (l + w)$ , where $P$ is the perimeter, $l$ is the length, and $w$ is the width.

STEP 4

Substitute the known values: $30 = 2 \cdot (9 + w)$ .

STEP 5

Divide both sides by $2$ to simplify: $15 = 9 + w$ .
We're doing this to isolate $w$ .

STEP 6

Subtract $9$ from both sides: $15 - 9 = w$ , so the reduced width is $w = 6$ meters.
We're subtracting $9$ to get $w$ all by itself!

STEP 7

We need to figure out how much smaller the reduced rectangle is compared to the original.
We can do this by dividing the reduced length by the original length.
Oh wait, we don't know the original length yet!
Don't worry, we'll find it soon!

STEP 8

Let's use the fact that the original perimeter is $120$ meters.
We also know that the perimeter of the reduced rectangle is $30$ meters.
The ratio of the perimeters is the same as the ratio of the lengths and widths.

STEP 9

Calculate the ratio of the original perimeter to the reduced perimeter: $120 / 30 = 4$ .
This tells us that the original rectangle is 4 times bigger than the reduced one.
This is our scaling factor!

STEP 10

Now we can find the original length! Multiply the reduced length by the scaling factor: $9 \cdot 4 = 36$ meters.
So, the original length is 36 meters!

STEP 11

We can use the same scaling factor to find the original width. Multiply the reduced width by the scaling factor: $6 \cdot 4 = 24$ meters.

STEP 12

Alternatively, we can use the original perimeter and length to find the width.
We know the original perimeter is $120$ meters and the original length is $36$ meters. Substitute these values into the perimeter formula: $120 = 2 \cdot (36 + w)$ .

STEP 13

Divide both sides by $2$ : $60 = 36 + w$ .

STEP 14

Subtract $36$ from both sides: $60 - 36 = w$ , so $w = 24$ meters.
Awesome, we got the same answer!

STEP 15

The width of the original rectangle is 24 meters.

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Studdy Solution

STEP 1

What is this asking? We're looking for the original rectangle's width, knowing its perimeter and the smaller rectangle's perimeter and length. Watch out! Don't mix up the original and reduced rectangle dimensions!

STEP 2

1. Find the reduced width
2. Find the reduced scale
3. Find the original length
4. Calculate the original width

STEP 3

We know the reduced rectangle's perimeter is $30$ meters and its length is $9$ meters.
Let's use the perimeter formula: $P = 2 \cdot (l + w)$ , where $P$ is the perimeter, $l$ is the length, and $w$ is the width.

STEP 4

Substitute the known values: $30 = 2 \cdot (9 + w)$ .

STEP 5

Divide both sides by $2$ to simplify: $15 = 9 + w$ .
We're doing this to isolate $w$ .

STEP 6

Subtract $9$ from both sides: $15 - 9 = w$ , so the reduced width is $w = 6$ meters.
We're subtracting $9$ to get $w$ all by itself!

STEP 7

We need to figure out how much smaller the reduced rectangle is compared to the original.
We can do this by dividing the reduced length by the original length.
Oh wait, we don't know the original length yet!
Don't worry, we'll find it soon!

STEP 8

Let's use the fact that the original perimeter is $120$ meters.
We also know that the perimeter of the reduced rectangle is $30$ meters.
The ratio of the perimeters is the same as the ratio of the lengths and widths.

STEP 9

Calculate the ratio of the original perimeter to the reduced perimeter: $120 / 30 = 4$ .
This tells us that the original rectangle is 4 times bigger than the reduced one.
This is our scaling factor!

STEP 10

Now we can find the original length! Multiply the reduced length by the scaling factor: $9 \cdot 4 = 36$ meters.
So, the original length is 36 meters!

STEP 11

We can use the same scaling factor to find the original width. Multiply the reduced width by the scaling factor: $6 \cdot 4 = 24$ meters.

STEP 12

Alternatively, we can use the original perimeter and length to find the width.
We know the original perimeter is $120$ meters and the original length is $36$ meters. Substitute these values into the perimeter formula: $120 = 2 \cdot (36 + w)$ .

STEP 13

Divide both sides by $2$ : $60 = 36 + w$ .

STEP 14

Subtract $36$ from both sides: $60 - 36 = w$ , so $w = 24$ meters.
Awesome, we got the same answer!

STEP 15

The width of the original rectangle is 24 meters.

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Studdy Solution

STEP 1

What is this asking? We're looking for the original rectangle's width, knowing its perimeter and the smaller rectangle's perimeter and length. Watch out! Don't mix up the original and reduced rectangle dimensions!

STEP 2

1. Find the reduced width2. Find the reduced scale3. Find the original length4. Calculate the original width

STEP 3

We know the reduced rectangle's perimeter is 303030 meters and its length is 999 meters.Let's use the perimeter formula: P=2⋅(l+w)P = 2 \cdot (l + w)P=2⋅(l+w), where PPP is the perimeter, lll is the length, and www is the width.

STEP 4

**Substitute** the **known values**: 30=2⋅(9+w)30 = 2 \cdot (9 + w)30=2⋅(9+w).

STEP 5

**Divide both sides** by 222 to simplify: 15=9+w15 = 9 + w15=9+w.We're doing this to isolate www.

STEP 6

**Subtract** 999 from both sides: 15−9=w15 - 9 = w15−9=w, so the reduced width is w=6w = 6w=6 meters.We're subtracting 999 to get www all by itself!

STEP 7

We need to figure out how much smaller the reduced rectangle is compared to the original.We can do this by dividing the reduced length by the original length.Oh wait, we don't know the original length yet!Don't worry, we'll find it soon!

STEP 8

Let's use the fact that the original perimeter is 120120120 meters.We also know that the perimeter of the reduced rectangle is 303030 meters.The ratio of the perimeters is the same as the ratio of the lengths and widths.

STEP 9

**Calculate** the ratio of the original perimeter to the reduced perimeter: 120/30=4120 / 30 = 4120/30=4.This tells us that the original rectangle is **4 times bigger** than the reduced one.This is our **scaling factor**!

STEP 10

Now we can find the original length! **Multiply** the reduced length by the scaling factor: 9⋅4=369 \cdot 4 = 369⋅4=36 meters.So, the original length is **36 meters**!

STEP 11

We can use the same scaling factor to find the original width. **Multiply** the reduced width by the scaling factor: 6⋅4=246 \cdot 4 = 246⋅4=24 meters.

STEP 12

Alternatively, we can use the original perimeter and length to find the width.We know the original perimeter is 120120120 meters and the original length is 363636 meters. **Substitute** these values into the perimeter formula: 120=2⋅(36+w)120 = 2 \cdot (36 + w)120=2⋅(36+w).

STEP 13

**Divide both sides** by 222: 60=36+w60 = 36 + w60=36+w.

STEP 14

**Subtract** 363636 from both sides: 60−36=w60 - 36 = w60−36=w, so w=24w = 24w=24 meters.Awesome, we got the same answer!

STEP 15

The width of the original rectangle is **24 meters**.

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1. Find the reduced width
2. Find the reduced scale
3. Find the original length
4. Calculate the original width

We know the reduced rectangle's perimeter is $30$ meters and its length is $9$ meters.
Let's use the perimeter formula: $P = 2 \cdot (l + w)$ , where $P$ is the perimeter, $l$ is the length, and $w$ is the width.

Substitute the known values: $30 = 2 \cdot (9 + w)$ .

Divide both sides by $2$ to simplify: $15 = 9 + w$ .
We're doing this to isolate $w$ .

Subtract $9$ from both sides: $15 - 9 = w$ , so the reduced width is $w = 6$ meters.
We're subtracting $9$ to get $w$ all by itself!

We need to figure out how much smaller the reduced rectangle is compared to the original.
We can do this by dividing the reduced length by the original length.
Oh wait, we don't know the original length yet!
Don't worry, we'll find it soon!

Let's use the fact that the original perimeter is $120$ meters.
We also know that the perimeter of the reduced rectangle is $30$ meters.
The ratio of the perimeters is the same as the ratio of the lengths and widths.

Calculate the ratio of the original perimeter to the reduced perimeter: $120 / 30 = 4$ .
This tells us that the original rectangle is 4 times bigger than the reduced one.
This is our scaling factor!

Now we can find the original length! Multiply the reduced length by the scaling factor: $9 \cdot 4 = 36$ meters.
So, the original length is 36 meters!

We can use the same scaling factor to find the original width. Multiply the reduced width by the scaling factor: $6 \cdot 4 = 24$ meters.

Alternatively, we can use the original perimeter and length to find the width.
We know the original perimeter is $120$ meters and the original length is $36$ meters. Substitute these values into the perimeter formula: $120 = 2 \cdot (36 + w)$ .

Divide both sides by $2$ : $60 = 36 + w$ .

Subtract $36$ from both sides: $60 - 36 = w$ , so $w = 24$ meters.
Awesome, we got the same answer!

The width of the original rectangle is 24 meters.