Mark has made a clay model.
He will now make a clay statue that is mathematically similar to the clay model.
The model has a base area if 6cm^2.
The statue will have a base area of 253.5cm^2.
Mark used 2kg of clay to make the model.
Clay is sold in 10kg bags.
Mark has to but all the clay he needs to make the statue.
How many bags of clay will Mark need to buy?
Thank you.
The two areas of similar objects are proportional to the square of their corresponding sides.
The two volumes of similar objects are proportional to the cubes of their corresponding sides.
sides of model^2 : sides of statue^2 = 6 : 253.5
side of model : side of statue = √6 : √253.5
Let's assume that the volume is proportional to the mass, let the mass of the statue be M
2 : M = (√6)^3 : (√253.5)^3
2(√6)^3 M = (√253.5)^3
M = (√253.5)^3 / 2(√6)^3
= 274.625 kg
So he needs 274.625/10 bags of clay
= 27.46 bags
So he must buy 28 bags of clay
He will need
(253.5/6)^(3/2) * 2 = 549.25 kg of clay
So, he will need 55 10kg bags
To find out how many bags of clay Mark needs to buy, we first need to determine the ratio of the base areas of the model and the statue.
The area of the model's base is 6 cm^2.
The area of the statue's base is 253.5 cm^2.
The ratio of the base areas is:
Ratio = Area of statue's base / Area of model's base
Ratio = 253.5 cm^2 / 6 cm^2
Ratio = 253.5 / 6
Ratio = 42.25
Since the statue is mathematically similar to the model, the ratio of their base areas is also the same as the ratio of their volumes.
We know that the model used 2 kg of clay. Let's assume that the statue will require x kg of clay.
The ratio of the base areas is the same as the ratio of the volumes:
Ratio = Volume of statue / Volume of model
42.25 = x kg / 2 kg
To find x, we can cross multiply:
42.25 * 2 = x
84.5 = x
So, Mark will need 84.5 kg of clay to make the statue.
Since clay is sold in 10 kg bags, we can now determine how many bags Mark needs to buy:
Number of bags = Total clay needed / Amount of clay in each bag
Number of bags = 84.5 kg / 10 kg
Number of bags = 8.45
Since we cannot buy a fraction of a bag, Mark will need to buy 9 bags of clay.
Therefore, Mark needs to buy 9 bags of clay to make the statue.
To solve this problem, we need to determine the amount of clay (in kilograms) required to make the statue, and then calculate the number of clay bags Mark needs to purchase.
The clay model and the clay statue are mathematically similar, which means their dimensions are proportional. Since the base area of the model is 6 cm^2 and the base area of the statue is 253.5 cm^2, we can find the ratio between the two base areas:
Ratio of base areas = (base area of statue) / (base area of model)
= 253.5 cm^2 / 6 cm^2
= 42.25
This ratio can be applied to the other dimensions of the clay model and statue as well.
Now, let's assume that the height of the model is h cm. To find the height of the statue, we can multiply the ratio of the base areas by the height of the model:
Height of statue = (ratio of base areas) * (height of model)
= 42.25 * h
= 42.25h cm
So, the statue will have a height of 42.25h cm.
Next, we need to determine the amount of clay required to make the statue. Since clay is three-dimensional, we need to consider the volume. The volume of a clay model is proportional to the cube of its dimensions, and the ratio between the model and statue dimensions is the same as the ratio between the base areas. Therefore, we have:
Ratio of volumes = (height of statue) / (height of model)
= (42.25h cm) / h
= 42.25
We know that Mark used 2 kg of clay to make the model. Let's assume the amount of clay required to make the statue is C kg. Using the ratio of volumes, we can set up the following proportion:
C / 2 kg = 42.25
To find C, we can solve for it:
C = 2 kg * 42.25
C = 84.5 kg
Therefore, Mark needs 84.5 kg of clay to make the statue.
Now, let's calculate the number of bags Mark needs to purchase. Since each bag contains 10 kg of clay, we divide the total amount of clay required by the weight of each bag:
Number of bags = (total amount of clay required) / (weight of each bag)
= 84.5 kg / 10 kg
= 8.45
Since we can't purchase a fractional number of bags, Mark will need to buy 9 bags of clay.
In summary, Mark needs to buy 9 bags of clay to make the clay statue.