Math / Geometry

QuestionPart III ${ }^{3}$ The students have planted a semicircle flower garden that requires some fencing a) Find the amount of fencing, to the nearest foot, required to fence this garden (Remember: You will need to find half of the circumference + the diameter to get the full perimeter) The amount of fencing that will be needed to fence the garden is b) If the fencing company charges $\$ 22$ per foot for fencing, how much would it cost in total to fence this flower garden? \ $22 per foot for fencing would be \$ c) Find the area of this flower garden. If a bag of mulch covers$ 10 \mathrm{ft}^{2}$, how many bags of mulch will we need? We will need
Part IV: There is a large circle painted in the middle of the park. The circle is divided into 4 equal sections. The students are going to paint each section a different color. The radius of the circle is 4 meters. a) What is the area of the entire circle?
The area of the entire circle is b) What is the area of each section of the circle?
The area of each section in the circle is c) After painting three sections, the following paint amount is left over. Which paint color has enough left to paint the last section? The paint color that has enough paint left to paint the last section is
Enough for $10 \mathrm{~m}^{2}$
Enough for $15 \mathrm{~m}^{2}$ Page 2

Studdy Solution

STEP 1

What is this asking? We need to figure out the perimeter, cost, and area of a semi-circular garden, and then deal with the area and paint needs of a circular park area divided into four sections. Watch out! Don't forget that the perimeter of a semicircle includes the straight edge (diameter) too!
Also, make sure to round up when calculating the number of mulch bags, since we can't buy fractions of bags.

STEP 2

1. Garden Fencing
2. Garden Cost
3. Garden Area and Mulch
4. Park Area
5. Section Area
6. Paint Check

STEP 3

Alright class, let's start with the garden!
It's a semicircle, so its perimeter is half the circumference of a full circle plus the diameter.
The radius is given as $\text{10 ft}$ .

STEP 4

The circumference of a full circle is $2 \cdot \pi \cdot r$ , so half of that is just $\pi \cdot r$ .
With $r = \text{10 ft}$ , half the circumference is $\pi \cdot \text{10 ft} \approx \text{31.416 ft}$ .

STEP 5

The diameter is twice the radius, so it's $2 \cdot \text{10 ft} = \text{20 ft}$ .

STEP 6

Adding those together, the total perimeter is roughly $\text{31.416 ft} + \text{20 ft} = \text{51.416 ft}$ .
Rounding to the nearest foot, we get $\text{51 ft}$ .

STEP 7

Each foot of fencing costs $\$22$ , and we need about $\text{51 ft}$ .

STEP 8

So, the total cost is $\$22/\text{ft} \cdot \text{51 ft} = \$1122$ .

STEP 9

The area of a full circle is $\pi \cdot r^2$ , so the area of our semicircle is half of that: $(\pi \cdot r^2)/2$ .
With $r = \text{10 ft}$ , the area is $(\pi \cdot (\text{10 ft})^2)/2 \approx \text{157.08 ft}^2$ .

STEP 10

Each bag of mulch covers $\text{10 ft}^2$ .
So, we need $\text{157.08 ft}^2 / \text{10 ft}^2/\text{bag} \approx \text{15.708 bags}$ .

STEP 11

Since we can't buy parts of bags, we'll need to round up to 16 bags.

STEP 12

Now, onto the park!
The circle has a radius of $\text{4 m}$ .
The area of a circle is $\pi \cdot r^2$ , so the park's area is $\pi \cdot (\text{4 m})^2 = \text{16}\pi \text{ m}^2 \approx \text{50.265 m}^2$ .

STEP 13

The circle is divided into four equal sections.

STEP 14

So, the area of each section is the total area divided by 4: $\text{50.265 m}^2 / 4 \approx \text{12.566 m}^2$ .

STEP 15

We need to figure out which paint color has enough left to cover the last section, which is about $\text{12.566 m}^2$ .

STEP 16

One color has enough for $\text{10 m}^2$ , which isn't enough.

STEP 17

The other color has enough for $\text{15 m}^2$ , which is enough!
So, that's the one we'll use.

STEP 18

a) The garden needs $\text{51 ft}$ of fencing. b) The fencing will cost $\$1122$ . c) We'll need $\text{16 bags}$ of mulch. d) The park's area is $\text{50.265 m}^2$ . e) Each section is $\text{12.566 m}^2$ . f) The paint that covers $\text{15 m}^2$ is sufficient for the last section.

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Garden Fencing
2. Garden Cost
3. Garden Area and Mulch
4. Park Area
5. Section Area
6. Paint Check

STEP 3

Alright class, let's start with the garden!
It's a semicircle, so its perimeter is half the circumference of a full circle plus the diameter.
The radius is given as $\text{10 ft}$ .

STEP 4

The circumference of a full circle is $2 \cdot \pi \cdot r$ , so half of that is just $\pi \cdot r$ .
With $r = \text{10 ft}$ , half the circumference is $\pi \cdot \text{10 ft} \approx \text{31.416 ft}$ .

STEP 5

The diameter is twice the radius, so it's $2 \cdot \text{10 ft} = \text{20 ft}$ .

STEP 6

Adding those together, the total perimeter is roughly $\text{31.416 ft} + \text{20 ft} = \text{51.416 ft}$ .
Rounding to the nearest foot, we get $\text{51 ft}$ .

STEP 7

Each foot of fencing costs $\$22$ , and we need about $\text{51 ft}$ .

STEP 8

So, the total cost is $\$22/\text{ft} \cdot \text{51 ft} = \$1122$ .

STEP 9

The area of a full circle is $\pi \cdot r^2$ , so the area of our semicircle is half of that: $(\pi \cdot r^2)/2$ .
With $r = \text{10 ft}$ , the area is $(\pi \cdot (\text{10 ft})^2)/2 \approx \text{157.08 ft}^2$ .

STEP 10

Each bag of mulch covers $\text{10 ft}^2$ .
So, we need $\text{157.08 ft}^2 / \text{10 ft}^2/\text{bag} \approx \text{15.708 bags}$ .

STEP 11

Since we can't buy parts of bags, we'll need to round up to 16 bags.

STEP 12

Now, onto the park!
The circle has a radius of $\text{4 m}$ .
The area of a circle is $\pi \cdot r^2$ , so the park's area is $\pi \cdot (\text{4 m})^2 = \text{16}\pi \text{ m}^2 \approx \text{50.265 m}^2$ .

STEP 13

The circle is divided into four equal sections.

STEP 14

So, the area of each section is the total area divided by 4: $\text{50.265 m}^2 / 4 \approx \text{12.566 m}^2$ .

STEP 15

We need to figure out which paint color has enough left to cover the last section, which is about $\text{12.566 m}^2$ .

STEP 16

One color has enough for $\text{10 m}^2$ , which isn't enough.

STEP 17

The other color has enough for $\text{15 m}^2$ , which is enough!
So, that's the one we'll use.

STEP 18

a) The garden needs $\text{51 ft}$ of fencing. b) The fencing will cost $\$1122$ . c) We'll need $\text{16 bags}$ of mulch. d) The park's area is $\text{50.265 m}^2$ . e) Each section is $\text{12.566 m}^2$ . f) The paint that covers $\text{15 m}^2$ is sufficient for the last section.

Get Studdy

Studdy solves anything!

Start learning now

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

Studdy Solution

STEP 1

STEP 2

1. Garden Fencing2. Garden Cost3. Garden Area and Mulch4. Park Area5. Section Area6. Paint Check

STEP 3

Alright class, let's **start** with the garden!It's a semicircle, so its perimeter is half the circumference of a full circle *plus* the diameter.The radius is given as 10 ft\text{10 ft}10 ft.

STEP 4

The **circumference** of a full circle is 2⋅π⋅r2 \cdot \pi \cdot r2⋅π⋅r, so half of that is just π⋅r\pi \cdot rπ⋅r.With r=10 ftr = \text{10 ft}r=10 ft, half the circumference is π⋅10 ft≈31.416 ft\pi \cdot \text{10 ft} \approx \text{31.416 ft}π⋅10 ft≈31.416 ft.

STEP 5

The **diameter** is twice the radius, so it's 2⋅10 ft=20 ft2 \cdot \text{10 ft} = \text{20 ft}2⋅10 ft=20 ft.

STEP 6

Adding those together, the **total perimeter** is roughly 31.416 ft+20 ft=51.416 ft\text{31.416 ft} + \text{20 ft} = \text{51.416 ft}31.416 ft+20 ft=51.416 ft.Rounding to the nearest foot, we get 51 ft\text{51 ft}51 ft.

STEP 7

Each foot of fencing costs $22\$22$22, and we need about 51 ft\text{51 ft}51 ft.

STEP 8

So, the **total cost** is $22/ft⋅51 ft=$1122\$22/\text{ft} \cdot \text{51 ft} = \$1122$22/ft⋅51 ft=$1122.

STEP 9

STEP 10

Each bag of mulch covers 10 ft2\text{10 ft}^210 ft2.So, we need 157.08 ft2/10 ft2/bag≈15.708 bags\text{157.08 ft}^2 / \text{10 ft}^2/\text{bag} \approx \text{15.708 bags}157.08 ft2/10 ft2/bag≈15.708 bags.

STEP 11

Since we can't buy parts of bags, we'll need to round up to **16 bags**.

STEP 12

Now, onto the park!The circle has a radius of 4 m\text{4 m}4 m.The **area** of a circle is π⋅r2\pi \cdot r^2π⋅r2, so the park's area is π⋅(4 m)2=16π m2≈50.265 m2\pi \cdot (\text{4 m})^2 = \text{16}\pi \text{ m}^2 \approx \text{50.265 m}^2π⋅(4 m)2=16π m2≈50.265 m2.

STEP 13

The circle is divided into **four equal sections**.

STEP 14

So, the **area of each section** is the total area divided by 4: 50.265 m2/4≈12.566 m2\text{50.265 m}^2 / 4 \approx \text{12.566 m}^250.265 m2/4≈12.566 m2.

STEP 15

We need to figure out which paint color has enough left to cover the last section, which is about 12.566 m2\text{12.566 m}^212.566 m2.

STEP 16

One color has enough for 10 m2\text{10 m}^210 m2, which isn't enough.

STEP 17

The other color has enough for 15 m2\text{15 m}^215 m2, which *is* enough!So, that's the one we'll use.

STEP 18

Get Studdy

Studdy solves anything!

Start learning now

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

1. Garden Fencing
2. Garden Cost
3. Garden Area and Mulch
4. Park Area
5. Section Area
6. Paint Check

Alright class, let's start with the garden!
It's a semicircle, so its perimeter is half the circumference of a full circle plus the diameter.
The radius is given as $\text{10 ft}$ .

The circumference of a full circle is $2 \cdot \pi \cdot r$ , so half of that is just $\pi \cdot r$ .
With $r = \text{10 ft}$ , half the circumference is $\pi \cdot \text{10 ft} \approx \text{31.416 ft}$ .

The diameter is twice the radius, so it's $2 \cdot \text{10 ft} = \text{20 ft}$ .

Adding those together, the total perimeter is roughly $\text{31.416 ft} + \text{20 ft} = \text{51.416 ft}$ .
Rounding to the nearest foot, we get $\text{51 ft}$ .

Each foot of fencing costs $\$22$ , and we need about $\text{51 ft}$ .

So, the total cost is $\$22/\text{ft} \cdot \text{51 ft} = \$1122$ .

Each bag of mulch covers $\text{10 ft}^2$ .
So, we need $\text{157.08 ft}^2 / \text{10 ft}^2/\text{bag} \approx \text{15.708 bags}$ .

Since we can't buy parts of bags, we'll need to round up to 16 bags.

Now, onto the park!
The circle has a radius of $\text{4 m}$ .
The area of a circle is $\pi \cdot r^2$ , so the park's area is $\pi \cdot (\text{4 m})^2 = \text{16}\pi \text{ m}^2 \approx \text{50.265 m}^2$ .

The circle is divided into four equal sections.

So, the area of each section is the total area divided by 4: $\text{50.265 m}^2 / 4 \approx \text{12.566 m}^2$ .

We need to figure out which paint color has enough left to cover the last section, which is about $\text{12.566 m}^2$ .

One color has enough for $\text{10 m}^2$ , which isn't enough.

The other color has enough for $\text{15 m}^2$ , which is enough!
So, that's the one we'll use.