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 larger cones, find how many of the smaller cones can be painted with the same amount of paint. (c) If the same amount of paint covers 24 similar cones, find the radius of one of these cones. Need help with part c .Thanks
I assume you got
(a) r = 20
(b) n = 600
(c) total volume of paint for 216 large cones is 216 * 400π/3
divide that by 24 and you get 3600 π/3 = π/3 * 60^2
so, r=60
Yeah, I know there's a factor of h involved in the area, but it is everywhere, and cancels out. So I just assumed h=1 to make things simpler.
thx dude
To find the radius of one of the cones in part (c), we can use the information given in parts (a) and (b).
In part (a), we are told that the ratio of the corresponding lengths of the two cones is 3:5. Since the length of a cone is proportional to its radius, we can set up the following ratio:
(radius of smaller cone) / (radius of larger cone) = 3/5
Substituting the value given in part (a), which is the radius of the smaller cone = 12 cm:
12 cm / (radius of larger cone) = 3/5
To solve for the radius of the larger cone, we can cross-multiply and solve the resulting equation:
5 * 12 cm = 3 * (radius of larger cone)
60 cm = 3 * (radius of larger cone)
Divide both sides of the equation by 3:
60 cm / 3 = (radius of larger cone)
20 cm = (radius of larger cone)
Therefore, the radius of the larger cone is 20 cm.
Moving on to part (b), we are given that there is sufficient paint to paint 216 larger cones. This means that the amount of paint needed to cover each larger cone is the same.
Next, we need to find out how many of the smaller cones can be painted with the same amount of paint. Since the paint covers the surface area of the cones, which is proportional to the square of the radius, we can set up the following ratio:
(number of smaller cones) / (number of larger cones) = (radius of smaller cone)^2 / (radius of larger cone)^2
Substituting the values given in parts (a) and (b):
(number of smaller cones) / 216 = (12 cm)^2 / (20 cm)^2
Simplifying the equation:
(number of smaller cones) / 216 = 144 cm^2 / 400 cm^2
Cross-multiplying and solving for the number of smaller cones:
(number of smaller cones) = (216 * 144 cm^2) / 400 cm^2
(number of smaller cones) = 7776 cm^2 / 400 cm^2
(number of smaller cones) ≈ 19.44
Since we can't have a fraction of a cone, we need to round 19.44 to the nearest whole number. Therefore, we can paint 19 smaller cones with the same amount of paint that can cover 216 larger cones.
I hope this helps! Let me know if you have any further questions.
To find the radius of one of the cones in part (c), we need to use the information given in part (b) as well as the fact that the cones are similar.
In part (b), we are told that there is sufficient paint to paint 216 larger cones. Since the cones are similar, the ratio of their volumes is equal to the ratio of the cubes of their corresponding lengths. Therefore, the ratio of the volume of a smaller cone to the volume of a larger cone is (3/5)^3.
Let the radius of one of the larger cones be r (in cm). We can set up the following equation based on the volumes of the cones:
(4/3)π(12^2)h / (4/3)π(r^2)h = (3/5)^3
Simplifying, we get:
(12^2) / (r^2) = (3/5)^3
To solve for r, we can take the square root of both sides:
r^2 = (12^2)/(3/5)^3
r^2 = (12^2)/(27/125)
r^2 = (12^2)*(125/27)
Taking the square root of both sides:
r = sqrt((12^2)*(125/27))
Using a calculator, we find:
r ≈ 17.88 cm
Therefore, the radius of one of the similar cones in part (c) is approximately 17.88 cm.