Two cones are similar in shape and the ratio of corresponding lengths is 3:5.
(a) if the radius of the smaller cone is 12cm, find the radius of the larger cone.
(b) if there is sufficient paint to paint 216 large cones, how many smaller cones can be painted with same amount of paint?
(c) If the same amount of paint covers 24 similar cones, what is the radius of one of these cones?
Question (a) is straight forward but I would really appreciate help with questions (b) and (c). My head is starting to hurt :-)
linear dimensions scale in the ratio of 3:5
areas scale in the ratio of (3:5)^2 = 9:25
Sure, let's solve these step-by-step:
(a) Since the two cones are similar, the ratio of their corresponding lengths is 3:5. The corresponding lengths of two cones include the radius as well. So, if the radius of the smaller cone is 12 cm, we can set up a proportion:
12 cm / x = 3/5
To solve for x (the radius of the larger cone), we can cross multiply:
3x = 5 * 12 cm
Now, we can solve for x:
3x = 60 cm
x = 60 cm / 3
x = 20 cm
Therefore, the radius of the larger cone is 20 cm.
(b) If there is sufficient paint to paint 216 large cones, we need to find out how many smaller cones can be painted with the same amount of paint. Since the volume of similar cones is proportional to the cube of their corresponding lengths, we can set up the following proportion:
(12 cm)^3 / (x cm)^3 = 216 / y
Where x is the radius of the smaller cone and y is the number of small cones.
Simplifying the equation:
(12^3) / (x^3) = 216 / y
1728 / x^3 = 216 / y
To find the number of smaller cones, we need to solve for y. Cross-multiplying gives:
1728y = 216x^3
y = (216x^3) / 1728
Since we know that 216 large cones can be painted, we can substitute that value into the equation:
y = (216 * 12^3) / 1728
Simplifying further:
y = 20736 / 1728
y = 12
Therefore, the same amount of paint can be used to paint 12 smaller cones.
(c) If the same amount of paint covers 24 similar cones, we need to find the radius of one of these cones.
Similar to the previous question, we can set up the following proportion:
(12 cm)^3 / (x cm)^3 = 24 / 1
Simplifying the equation:
1728 / x^3 = 24
To solve for x (the radius of one of the cones), we can cross-multiply:
24x^3 = 1728 cm^3
Now, we can solve for x:
x^3 = 1728 cm^3 / 24
x^3 = 72 cm^3
x = ∛(72 cm^3)
x = 4.21 cm (rounded to two decimal places)
Therefore, the radius of one of these cones is approximately 4.21 cm.
No worries! I'll guide you through the solutions for questions (b) and (c). Let's start by solving question (b) first.
To find out how many smaller cones can be painted with the same amount of paint that can paint 216 large cones, we need to understand the relationship between the volumes of the cones. The volume of a cone can be calculated using the formula:
Volume = (1/3) * π * r^2 * h
Since the two cones are similar, the ratio of their corresponding lengths is given as 3:5. This tells us that the ratio of their corresponding heights is also 3:5.
In question (b), we know that the number of large cones is 216. Let's denote the number of smaller cones as 'n'.
To find the number of smaller cones, we need to compare the volumes of the large and small cones. Since the ratio of the lengths is given as 3:5, the ratio of the volumes of the large and small cones will be (3/5)^3. This is because the volume of a cone is proportional to the cube of its radius.
So, we have:
(3/5)^3 = n/216
To solve for 'n', we can rearrange the equation:
n = 216 * (3/5)^3
Now, let's move on to question (c).
In question (c), we know that the same amount of paint covers 24 similar cones. We need to find the radius of one of these cones.
Again, since the cones are similar, the ratio of their corresponding volumes is the cube of the ratio of their corresponding lengths. We can use this information to find the ratio of the radii.
Let's denote the radius of one of the cones as 'r'. Now, comparing the volumes, we have:
24 large cone volume : 1 small cone volume = (3/5)^3
The volume of a cone is calculated as (1/3) * π * r^2 * h, and since the height of the cones is not given, we can disregard it. Thus, the ratio of the volumes is equal to the ratio of the radii squared.
So, we have:
24 * (π * r_large^2) : (π * r_small^2) = (3/5)^3
Simplifying this equation, we get:
24 * r_large^2 : r_small^2 = (3/5)^3
Since the ratio of the radii is squared, we can rewrite the equation as:
24 * r_large^2 / r_small^2 = (3/5)^3
To solve for the radius of one of the cones ('r'), we can rearrange the equation as:
r = √[(24 * r_large^2) / ((3/5)^3)]
I hope this explanation helps! Let me know if you need further clarification.