Math / Geometry

QuestionSolving Problems with the Area of a Circle Answer the questions below based on the diagrams given. Use 3.14 as an approximation for $\pi$ in all of your computations. Round any answers to two decimal places as needed.
Chloe and a friend are ordering pizza. They have a choice between a Medium or Large Pizza as shown in the diagrams above. How many square inches is the Medium Pizza? How many square inches is the Large Pizza? in $^{2}$ $\square$ in $^{2}$
The diameter of the Large pizza is $\square$ times as large as the Medium Pizza.
The area of the Large pizza is $\square$ times as large as the Medium Pizza.
If the Medium Pizza costs $\$ 11$ , how much should the Large pizza cost to be of equal value? $\$$ $\square$ Pizza Math Moral: Unless you have a coupon or there is a special deal going on, it is almost always a better value to order the larger pizza!

Studdy Solution

STEP 1

What is this asking? We need to figure out the area of two different sized pizzas, compare their sizes and prices, and see how their values stack up! Watch out! Don't mix up radius and diameter!
Remember the radius is half the diameter.

STEP 2

1. Calculate the area of the medium pizza.
2. Calculate the area of the large pizza.
3. Compare the diameters.
4. Compare the areas.
5. Determine the proportional price of the large pizza.

STEP 3

The diameter of the medium pizza is $14$ inches.
The radius is half the diameter, so the radius is $14 \div 2 = 7$ inches.
We'll need this for the area formula!

STEP 4

The area of a circle is $\pi \cdot r^2$ , where $r$ is the radius.
Using our radius of $7$ inches and $\pi \approx 3.14$ , the area of the medium pizza is $3.14 \cdot 7^2 = 3.14 \cdot 49 = 153.86$ square inches.

STEP 5

The large pizza has a diameter of $16$ inches, meaning its radius is $16 \div 2 = 8$ inches.

STEP 6

Using the same formula as before, the area of the large pizza is $3.14 \cdot 8^2 = 3.14 \cdot 64 = 200.96$ square inches.

STEP 7

The large pizza's diameter is $16$ inches and the medium pizza's diameter is $14$ inches.
To find how many times larger the large diameter is, we divide: $16 \div 14 = \frac{16}{14} = \frac{8}{7} \approx 1.14$ .

STEP 8

So, the large pizza's diameter is approximately 1.14 times larger than the medium pizza's diameter.

STEP 9

We found the large pizza's area is $200.96$ square inches, and the medium pizza's area is $153.86$ square inches.
The ratio of the areas is $200.96 \div 153.86 = \frac{200.96}{153.86} \approx 1.31$ .

STEP 10

The large pizza has about 1.31 times the area of the medium pizza.

STEP 11

If the medium pizza costs $\$11$ , and the large pizza is $1.31$ times bigger, then a fair price for the large pizza would be $11 \cdot 1.31 = \$14.41$ .
This keeps the price per square inch the same.

STEP 12

The medium pizza is $153.86$ square inches.
The large pizza is $200.96$ square inches.
The large pizza's diameter is approximately $1.14$ times larger than the medium's.
The large pizza's area is approximately $1.31$ times larger.
A proportional price for the large pizza would be $\$14.41$ .

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Studdy Solution

STEP 1

What is this asking? We need to figure out the area of two different sized pizzas, compare their sizes and prices, and see how their values stack up! Watch out! Don't mix up radius and diameter!
Remember the radius is half the diameter.

STEP 2

1. Calculate the area of the medium pizza.
2. Calculate the area of the large pizza.
3. Compare the diameters.
4. Compare the areas.
5. Determine the proportional price of the large pizza.

STEP 3

The diameter of the medium pizza is $14$ inches.
The radius is half the diameter, so the radius is $14 \div 2 = 7$ inches.
We'll need this for the area formula!

STEP 4

The area of a circle is $\pi \cdot r^2$ , where $r$ is the radius.
Using our radius of $7$ inches and $\pi \approx 3.14$ , the area of the medium pizza is $3.14 \cdot 7^2 = 3.14 \cdot 49 = 153.86$ square inches.

STEP 5

The large pizza has a diameter of $16$ inches, meaning its radius is $16 \div 2 = 8$ inches.

STEP 6

Using the same formula as before, the area of the large pizza is $3.14 \cdot 8^2 = 3.14 \cdot 64 = 200.96$ square inches.

STEP 7

The large pizza's diameter is $16$ inches and the medium pizza's diameter is $14$ inches.
To find how many times larger the large diameter is, we divide: $16 \div 14 = \frac{16}{14} = \frac{8}{7} \approx 1.14$ .

STEP 8

So, the large pizza's diameter is approximately 1.14 times larger than the medium pizza's diameter.

STEP 9

We found the large pizza's area is $200.96$ square inches, and the medium pizza's area is $153.86$ square inches.
The ratio of the areas is $200.96 \div 153.86 = \frac{200.96}{153.86} \approx 1.31$ .

STEP 10

The large pizza has about 1.31 times the area of the medium pizza.

STEP 11

If the medium pizza costs $\$11$ , and the large pizza is $1.31$ times bigger, then a fair price for the large pizza would be $11 \cdot 1.31 = \$14.41$ .
This keeps the price per square inch the same.

STEP 12

The medium pizza is $153.86$ square inches.
The large pizza is $200.96$ square inches.
The large pizza's diameter is approximately $1.14$ times larger than the medium's.
The large pizza's area is approximately $1.31$ times larger.
A proportional price for the large pizza would be $\$14.41$ .

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Studdy Solution

STEP 1

What is this asking? We need to figure out the area of two different sized pizzas, compare their sizes and prices, and see how their values stack up! Watch out! Don't mix up radius and diameter!Remember the radius is half the diameter.

STEP 2

1. Calculate the area of the medium pizza.2. Calculate the area of the large pizza.3. Compare the diameters.4. Compare the areas.5. Determine the proportional price of the large pizza.

STEP 3

The **diameter** of the medium pizza is 141414 inches.The radius is *half* the diameter, so the radius is 14÷2=714 \div 2 = 714÷2=7 inches.We'll need this for the area formula!

STEP 4

The area of a circle is π⋅r2\pi \cdot r^2π⋅r2, where rrr is the **radius**.Using our **radius** of 777 inches and π≈3.14\pi \approx 3.14π≈3.14, the area of the medium pizza is 3.14⋅72=3.14⋅49=153.863.14 \cdot 7^2 = 3.14 \cdot 49 = 153.863.14⋅72=3.14⋅49=153.86 **square inches**.

STEP 5

The large pizza has a **diameter** of 161616 inches, meaning its **radius** is 16÷2=816 \div 2 = 816÷2=8 inches.

STEP 6

Using the same formula as before, the area of the large pizza is 3.14⋅82=3.14⋅64=200.963.14 \cdot 8^2 = 3.14 \cdot 64 = 200.963.14⋅82=3.14⋅64=200.96 **square inches**.

STEP 7

The **large pizza's diameter** is 161616 inches and the **medium pizza's diameter** is 141414 inches.To find how many times larger the large diameter is, we divide: 16÷14=1614=87≈1.1416 \div 14 = \frac{16}{14} = \frac{8}{7} \approx 1.1416÷14=1416​=78​≈1.14.

STEP 8

So, the **large pizza's diameter** is approximately **1.14 times** larger than the **medium pizza's diameter**.

STEP 9

We found the **large pizza's area** is 200.96200.96200.96 square inches, and the **medium pizza's area** is 153.86153.86153.86 square inches.The ratio of the areas is 200.96÷153.86=200.96153.86≈1.31200.96 \div 153.86 = \frac{200.96}{153.86} \approx 1.31200.96÷153.86=153.86200.96​≈1.31.

STEP 10

The **large pizza** has about **1.31 times** the area of the **medium pizza**.

STEP 11

If the **medium pizza** costs $11\$11$11, and the **large pizza** is 1.311.311.31 times bigger, then a fair price for the **large pizza** would be 11⋅1.31=$14.4111 \cdot 1.31 = \$14.4111⋅1.31=$14.41.This keeps the price per square inch the same.

STEP 12

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What is this asking? We need to figure out the area of two different sized pizzas, compare their sizes and prices, and see how their values stack up! Watch out! Don't mix up radius and diameter!
Remember the radius is half the diameter.

1. Calculate the area of the medium pizza.
2. Calculate the area of the large pizza.
3. Compare the diameters.
4. Compare the areas.
5. Determine the proportional price of the large pizza.

The diameter of the medium pizza is $14$ inches.
The radius is half the diameter, so the radius is $14 \div 2 = 7$ inches.
We'll need this for the area formula!

The area of a circle is $\pi \cdot r^2$ , where $r$ is the radius.
Using our radius of $7$ inches and $\pi \approx 3.14$ , the area of the medium pizza is $3.14 \cdot 7^2 = 3.14 \cdot 49 = 153.86$ square inches.

The large pizza has a diameter of $16$ inches, meaning its radius is $16 \div 2 = 8$ inches.

Using the same formula as before, the area of the large pizza is $3.14 \cdot 8^2 = 3.14 \cdot 64 = 200.96$ square inches.

The large pizza's diameter is $16$ inches and the medium pizza's diameter is $14$ inches.
To find how many times larger the large diameter is, we divide: $16 \div 14 = \frac{16}{14} = \frac{8}{7} \approx 1.14$ .

So, the large pizza's diameter is approximately 1.14 times larger than the medium pizza's diameter.

We found the large pizza's area is $200.96$ square inches, and the medium pizza's area is $153.86$ square inches.
The ratio of the areas is $200.96 \div 153.86 = \frac{200.96}{153.86} \approx 1.31$ .

The large pizza has about 1.31 times the area of the medium pizza.

If the medium pizza costs $\$11$ , and the large pizza is $1.31$ times bigger, then a fair price for the large pizza would be $11 \cdot 1.31 = \$14.41$ .
This keeps the price per square inch the same.