3x+y=9 in slope intercept and point slope form
1 answer:
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Option A: The height of the cylinder is equal to the diameter of the sphere.
Option C: The radius of the sphere is half the height of the cylinder.
Option E: The volume of the sphere is two-thirds the volume of the cylinder.
Solution:
The sphere is inside the cylinder.
Let r be the radius of the sphere.
Option A: The height of the cylinder is equal to the diameter of the sphere.
The sphere is fully occupied the cylinder.
If we draw the vertical line through enter of the sphere, which is equal to the height of cylinder.That is h = d. It is true.
Option B: The height of the cylinder is two times the diameter of the sphere.
That is h = 2d. From the above option, we know that h = d.
So, it is not true.
Option C: The radius of the sphere is half the height of the cylinder.
we know that diameter = 2 × radius (d = 2r)
From option A, we have h = d.
Substitute d = 2r, we get
⇒ h = 2r
Divide by 2 on both sides, we get
Therefore, it is true.
Option D: The diameter of the sphere is equal to the radius of the cylinder.
It is not true, because diameters of both cylinder and sphere are equal.
Option E: The volume of the sphere is two-thirds the volume of the cylinder.
Radius of cylinder and sphere = r
Height of cylinder h = 2r (by option C)
Volume of cylinder = Volume of sphere
(formula)
Hence it is true.
Option A, option C and option E are true.
I have attached a screenshot with the images of the two popcorn bags used by the theater.
Part (a):
Bag A:
Volume of bag = area of base * height
Volume of bag = length * width * height
We have:
volume = 96 in³
length = 3 in
width = 4 in
Therefore:
96 = 3 * 4 * height
96 = 12 * height
height of bag A = 8 in
Bag B:
Volume of bag = area of base * height
Volume of bag = length * width * height
We have:
volume = 96 in³
length = 4 in
width = 4 in
Therefore:
96 = 4 * 4 * height
96 = 16 * height
height of bag B = 6 in
Part (b):
To determine the amount of paper needed, we will need to calculate the surface of each bag. Excluding the top base, each bag will have 5 faces. Four side faces (each two opposite are equal) and the base.
Bag A:
Surface area = area of base + 2*area of front face + 2*area of side face
Surface area = (3*4) + 2(8*4) + 2(8*3)
Surface area = 12 + 64 + 48
Surface area of bag A = 124 in²
Bag B:
Surface area = area of base + 2*area of front face + 2*area of side face
Surface area = (4*4) + 2(6*4) + 2(6*4)
Surface area = 16 + 48 + 48
Surface area of bag B = 112 in²
From the above calculations, we can deduce that the theater should choose bag B in order to reduce the amount of paper needed.
Hope this helps :)
Part (a):
Bag A:
Volume of bag = area of base * height
Volume of bag = length * width * height
We have:
volume = 96 in³
length = 3 in
width = 4 in
Therefore:
96 = 3 * 4 * height
96 = 12 * height
height of bag A = 8 in
Bag B:
Volume of bag = area of base * height
Volume of bag = length * width * height
We have:
volume = 96 in³
length = 4 in
width = 4 in
Therefore:
96 = 4 * 4 * height
96 = 16 * height
height of bag B = 6 in
Part (b):
To determine the amount of paper needed, we will need to calculate the surface of each bag. Excluding the top base, each bag will have 5 faces. Four side faces (each two opposite are equal) and the base.
Bag A:
Surface area = area of base + 2*area of front face + 2*area of side face
Surface area = (3*4) + 2(8*4) + 2(8*3)
Surface area = 12 + 64 + 48
Surface area of bag A = 124 in²
Bag B:
Surface area = area of base + 2*area of front face + 2*area of side face
Surface area = (4*4) + 2(6*4) + 2(6*4)
Surface area = 16 + 48 + 48
Surface area of bag B = 112 in²
From the above calculations, we can deduce that the theater should choose bag B in order to reduce the amount of paper needed.
Hope this helps :)