Cups A and B are cone shaped and have heights of #33 cm# and #37 cm# and openings with radii of #10 cm# and #7 cm#, respectively. If cup B is full and its contents are poured into cup A, will cup A overflow? If not how high will cup A be filled?

Answer 1

#color(blue)(h~~27.028" cm to 3 decimal places")#

#color(blue)("Test condition")#

Consider across section vertically through the centre of each cone.

If the area of the taller cross section will fit in the area of the shorter one then so will the volumes.

Let cross section area be #A#

we require that #A_B < A_A# ( area for #A_A# is bigger )

#"require that "7xx37< 10xx33#

#259 < 330# The test condition is true

#color(red)("It will fit!")#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("To find volume of cone B")#
Volume a circular cone is #1/3xx"base area"xx"height"#

Thus the volume of B is: #" "1/3 pi (7)^2 xx37cm^3#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("To find height within cone A")#

The radius will vary in accordance with the height from zero to full radius at the top height of 33cm.

So #color(green)("radius")# at any height is :# r/33xxh = color(green)(10/33xxh)#

The volume of the transferred material is#" "1/3 pi (7)^2 xx37cm^3#

So #cancel(1/3pi )(7)^2 xx37cm^3=cancel(1/3pi) (color(green)(10/33xxh))^2xx h#

#=>49xx37=color(green)(100/1089xxh^2)xxh#

#=>h= root(3)((49xx37xx1089)/100)#

#color(blue)(h~~27.028" cm to 3 decimal places")#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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Answer 2

height = #27.03cm#

We need to find and compare the Volumes of A and B. Check first whether they are similar in shape - this would make some of the calculations easier.

Are the sides in the same ratio? #7/10 =0.7 and 37/33 =1.12 " "rArr " A and B are not similar"#
#Vol_("cone") = (pi r^2 h)/3#
#Vol_A = (pi xx10^2xx33)/3" "and " "Vol_B = (pixx7^2xx37)/3#
Which is bigger? We do not need to include #pi and 3# in the comparison because they are common.
#color(white)(xxxxxxxxxx)10^2 xx33 > 7^2 xx37#

Therefore A will not overflow but we need to find the height.

The cone formed by the water in A and the whole cone of A are similar in shape.

The ratio of the cubes of the heights is equal to the ratio of the volumes.

#color(white)(xxxxxxxxxxxxxxxxx)h^3/H^3 = v/V#
#h^3/33^3 = (7^2 xx37)/(10^2 xx 33)#
#h^3 = (33^3 xx7^2 xx37)/(10^2xx33) = 19,743.57#
#h = root3(19,743.57)#
#27.03cm#
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Answer 3

No, cup A will not overflow. When the contents of cup B are poured into cup A, the volume of cup A will increase by the volume of cup B. Since the volume of a cone is given by V = (1/3)πr^2h, where r is the radius and h is the height, we can calculate the volumes of cups A and B and compare them.

Volume of cup A = (1/3)π(10^2)(33) ≈ 3465.98 cm^3 Volume of cup B = (1/3)π(7^2)(37) ≈ 3591.34 cm^3

The difference in volumes is approximately 125.36 cm^3. Since this is less than the volume of cup B, cup A will not overflow. Cup A will be filled to a height of 33 cm + (125.36 cm^3 / (π(10^2))) ≈ 36.41 cm.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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