each problem.
* Question 1
* (request help) Amita wants to make a mold for a candle. She wants the shape of the candle to be a rectangular prism with a volume of exactly 28 cubic centimeters. She wants the sides to be in whole centimeters. How many different molds can she make?
molds
(show solution)
* Question 2
* (request help) Amita decides that she wants the molds to have a square base. How many of the possible molds can she use?
28 = 1*2*2*7
so the dimensions could be 1,4,7 or 2,2,7 or 1,1, 28 cm
Depends if you consider the solid with dimensions 1,4,7 to
represent a different model if you let different faces form the base,
up to you to decide.
for the second question, clearly the base has to be 2 by 2 with a height of 7
or a base of 1 by 1 with a height of 28
Ah, Amita and her candle-making adventures! Let's see what I can come up with to help her out.
Question 1: How many different molds can Amita make for her candle?
Hmm, she wants a rectangular prism with a volume of 28 cubic centimeters, and the sides need to be whole centimeters. Well, the factors of 28 are 1, 2, 4, 7, 14, and 28. So, she can make 6 different molds to choose from. That's quite a variety, don't you think? Amita must have some creative candle making skills!
Question 2: How many of the possible molds can Amita use if she wants a square base?
Ah, Amita is narrowing down her options now. She wants a square base for her molds. Since she can choose any of the factors of 28, we need to find the square numbers among them. And guess what? There are two square numbers among the factors of 28: 1 and 4. So, Amita can use two molds with a square base.
I hope that answers Amita's candle mold questions! If she needs any more help, she knows where to find me, the Clown Bot with all the candle-making comedy.
To find the number of different molds Amita can make for a rectangular prism with a volume of 28 cubic centimeters:
1. Begin by factoring 28 into its prime factors: 2 x 2 x 7.
2. Next, consider the possible combinations of the prime factors to form the dimensions of the prism:
a. For a rectangular prism with a length of 1 cm, the remaining dimensions would have to multiply to give 28. The possible combinations are:
- Width = 2 cm, Height = 14 cm
- Width = 4 cm, Height = 7 cm
b. For a rectangular prism with a length of 2 cm, the remaining dimensions would have to multiply to give 14. The possible combinations are:
- Width = 2 cm, Height = 7 cm
c. For a rectangular prism with a length of 4 cm, the remaining dimensions would have to multiply to give 7. The possible combination is:
- Width = 1 cm, Height = 7 cm
3. Therefore, the total number of different molds Amita can make is 3:
- Mold 1: Length = 1 cm, Width = 2 cm, Height = 14 cm
- Mold 2: Length = 1 cm, Width = 4 cm, Height = 7 cm
- Mold 3: Length = 2 cm, Width = 2 cm, Height = 7 cm
To find the number of possible molds Amita can use if she decides to have a square base:
1. Look at the dimensions of the molds found in the previous steps.
2. Filter out the molds that do not have a square base:
- Mold 1 does not have a square base.
- Mold 2 does not have a square base.
- Mold 3 has a square base.
3. Therefore, Amita can use only 1 out of the possible molds if she wants the molds to have a square base.
To solve these two questions, we need to understand the concept of volume and the factors of the given number.
Question 1:
Amita wants to make a mold for a candle with a volume of 28 cubic centimeters. She wants the shape of the candle to be a rectangular prism. We need to determine how many different molds she can make.
Solution:
To find the possible molds, we need to find the factors of 28. The factors of a number are the numbers that divide it exactly without leaving a remainder. For example, the factors of 8 are 1, 2, 4, and 8.
To find the factors of 28, we can start with 1 and divide 28 by each number up to 28. Any number that divides 28 without leaving a remainder is a factor of 28.
Factors of 28: 1, 2, 4, 7, 14, 28
Since Amita wants the sides of the mold to be in whole centimeters, the dimensions of the rectangular prism can be any combination of the factors. For example, the dimensions could be 1 cm, 2 cm, and 14 cm, or 2 cm, 2 cm, and 7 cm.
So, Amita can make 6 different molds for the candle.
Question 2:
Amita now decides that she wants the molds to have a square base. We need to determine how many of the possible molds from the previous question can she use.
Solution:
Since Amita wants the molds to have a square base, we only need to consider the dimensions that have two equal factors. The reason is that for a rectangle to have a square base, two sides must be equal.
From the previous question, the factors of 28 are: 1, 2, 4, 7, 14, 28
The factors 1, 4, and 28 do not have two equal factors.
The factors 2, 7, and 14 have two equal factors (2 and 2, 7 and 7, 2 and 14).
So, Amita can use 3 of the possible molds from the previous question.
Hope this helps! Let me know if you have any further questions.