John planned a birthday party for his younger sister with his friends. They decided to make some birthday caps by themselves and to buy a cake from a bakery shop. For these two items, they decided the following dimensions :

Cake : Cylindrical shape with diameter 24 cm and height 14 cm.
Cap: Conical shape with base circumference 44 cm and height 24 cm. Based on the above information, answer the following questions :
(a) How many square cm paper would be used to make 4 such caps ?
(b)The bakery shop sells cakes by weight (0.5 kg, 1 kg, 1.5 kg, etc.). To have the required dimensions, how much cake should they order, if 650 cm^3 equals 100 g of cake?

cap: area = πrs

where r = 44/(2π) and s^2 = r^2 + h^2

cake: v = πr^2 h cm^3
weight = v * 100g/650cm^3 * 1kg/1000g

(a) Well, to calculate the total amount of square cm paper used to make 4 caps, we need to find the lateral surface area of one cap and then multiply it by 4.

The lateral surface area of a cone can be calculated using the formula: A = πrl, where r is the radius of the base and l is the slant height. We can find the radius using the given circumference of 44 cm.

C = 2πr
44 = 2πr
r = 22/π cm

Now, to find the slant height, we can use the Pythagorean theorem. The height (h) is given as 24 cm, and the radius (r) we just calculated.

l^2 = r^2 + h^2
l^2 = (22/π)^2 + 24^2
l ≈ 28.026 cm

So, the lateral surface area of one cone is A = π(22/π)(28.026) cm^2.

Now, to find the total area for 4 caps, we multiply this by 4:

Total Area = 4 × π(22/π)(28.026) cm^2
Total Area ≈ 2468.27 cm^2

So, approximately 2468.27 square cm of paper would be used to make 4 such caps.

(b) To find out how much cake they should order, we need to calculate the volume of the cylindrical cake.

The volume of a cylinder can be found using the formula: V = πr^2h, where r is the radius of the base and h is the height.

Given that the diameter is 24 cm, the radius (r) is half of that, which is 12 cm. The height (h) is given as 14 cm.

So, the volume of the cake is V = π(12)^2(14) cm^3.

Since we know that 650 cm^3 of cake equals 100 g, we can set up the following ratio:

650 cm^3 : 100 g = V cm^3 : x g

Cross-multiplying, we get:

x = (V × 100)/650 g

Plugging in the values, we get:

x = ((π(12)^2(14)) × 100)/650 g

Calculating this, you would have the amount of cake they should order in grams. But remember, this is just an approximate calculation.

P.S. Hope your sister's birthday turns out to be as sweet as the cake!

(a) To calculate the amount of paper required to make 4 caps, we need to find the lateral surface area of one cap and then multiply it by 4.

The lateral surface area of a conical cap can be calculated using the formula:
Lateral Surface Area = πrℓ, where r is the radius of the base and ℓ is the slant height.

First, we need to find the radius of the base using the formula:
Circumference = 2πr,
where the circumference is given as 44 cm.
Rearranging the formula to solve for r, we have:
r = Circumference / (2π)
= 44 cm / (2π)
≈ 7 cm

Next, we need to find the slant height (h) of the cap using the Pythagorean theorem:
h = √(r^2 + h^2),
where h is the height of the cap, given as 24 cm.

Substituting the values, we have:
h = √(7^2 + 24^2)
= √(49 + 576)
≈ √625
≈ 25 cm

Now, we can calculate the lateral surface area:
Lateral Surface Area = πrh
= π * 7 cm * 25 cm
≈ 550 cm^2

Therefore, the amount of paper required to make 4 such caps would be:
4 * 550 cm^2 = 2200 cm^2

(b) To calculate the amount of cake needed, we need to find the volume of the cake using the formula:
Volume = πr^2h, where r is the radius of the cake and h is the height.

Given the diameter of the cake is 24 cm, we can find the radius:
r = diameter / 2
= 24 cm / 2
= 12 cm

Now, we can calculate the volume:
Volume = π * (12 cm)^2 * 14 cm
≈ 2,985.24 cm^3

We know that 650 cm^3 of cake is equal to 100 g. To find how much cake we need, we can set up the following proportion:
650 cm^3 : 100 g = Volume : x g

Cross multiplying, we have:
650 cm^3 * x g = 100 g * Volume
x g = (100 g * Volume) / 650 cm^3
= (100 g * 2,985.24 cm^3) / 650 cm^3
= 460.03 g

Therefore, they should order approximately 460 g of cake.

To answer the questions, we need to calculate the surface area of the cap and the volume of the cake.

(a) To find the surface area of the cap, we need to use the formula for the lateral surface area of a cone, which is given by:

Surface area = π * r * l,

where r is the radius of the base and l is the slant height.

Given that the base circumference is 44 cm, we can find the radius (r) using the formula:

Circumference = 2 * π * r,

Simplifying the equation, we get:

r = Circumference / (2 * π) = 44 / (2 * 3.14) ≈ 7 cm.

Since the height is given as 24 cm, we can use the Pythagorean theorem to find the slant height (l):

l = sqrt(r^2 + h^2) = sqrt(7^2 + 24^2) ≈ 25.25 cm.

Now, let's calculate the surface area of the cap:

Surface area = π * r * l = 3.14 * 7 * 25.25 ≈ 553.25 cm^2.

So, to make 4 such caps, the total paper used would be:

Total paper used = Surface area * 4 = 553.25 * 4 ≈ 2213 cm^2.

Therefore, approximately 2213 square cm of paper would be used to make 4 caps.

(b) To find the volume of the cake, we need to use the formula for the volume of a cylinder, which is given by:

Volume = π * r^2 * h,

where r is the radius of the base and h is the height.

Given that the diameter is 24 cm, we can find the radius (r) by dividing the diameter by 2:

r = Diameter / 2 = 24 / 2 = 12 cm.

We are also given the height as 14 cm, so we can calculate the volume of the cake:

Volume = π * r^2 * h = 3.14 * 12^2 * 14 ≈ 7914.24 cm^3.

Now, we know that 650 cm^3 of cake weighs 100 g. So, let's calculate the weight of the cake needed:

Weight of cake = (Volume / 650) * 100 ≈ (7914.24 / 650) * 100 ≈ 1217 g.

Therefore, they should order approximately 1217 g of cake to have the required dimensions.