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A company uses waterproof paper to make disposable conical drinking cups - Leaving Cert Mathematics - Question 8 - 2012
Question 8
A company uses waterproof paper to make disposable conical drinking cups. To make each cup, a sector AOB is cut from a circular piece of paper of radius 9 cm. The ed... show full transcript
Worked Solution & Example Answer:A company uses waterproof paper to make disposable conical drinking cups - Leaving Cert Mathematics - Question 8 - 2012
Step 1
Show that the capacity of the cup, in cm³, is given by the formula V = \frac{\pi}{3}h(81 - h²)
Answer
To derive the formula for capacity, we start from the geometry of the conical cup. The relationship between the radius (r) and height (h) of the cup is r² + h² = 9², hus we can express r² as:
r² = 81 - h².
The volume V of a cone is given by the formula:
V = \frac{\pi}{3}r²h.
Substituting the expression for r² into the volume formula gives:
V = \frac{\pi}{3}(81 - h²)h,
= \frac{\pi}{3}h(81 - h²).
Step 2
Find the two values of h for which the capacity of the cup is \frac{154\pi}{3}
Answer
Setting the expression for volume equal to \frac{154\pi}{3}:
\frac{\pi}{3}h(81 - h²) = \frac{154\pi}{3}
We can simplify by canceling \frac{\pi}{3}:
h(81 - h²) = 154.
This expands to:
h² - 81h + 154 = 0.
Using the quadratic formula: h = \frac{81 \pm \sqrt{81² - 4 \cdot 1 \cdot 154}}{2 \cdot 1} h = \frac{81 \pm \sqrt{6561 - 616}}{2} h = \frac{81 \pm \sqrt{5945}}{2}.
The positive solutions will yield two values: an integer and a non-integer. They are h = 2 and h ≈ 7.83.
Step 3
Find the maximum possible volume of the cup, correct to the nearest cm³
Answer
To find the maximum volume, we take the derivative of V with respect to h:
V(h) = \frac{\pi}{3}h(81 - h²).
Thus, dV/dh = \frac{\pi}{3}(81 - 3h²).
Setting dV/dh = 0 for maximum volume gives:
81 - 3h² = 0 => h² = 27 => h = \sqrt{27} = 3\sqrt{3} ≈ 5.2 cm.
Substituting this value back into the volume formula yields V ≈ 294 cm³.
Step 4
Step 5
Step 6
Find the measure of the angle AOB that must be cut from the circular disc
Answer
The circumference of the rim is given by:
C = 2\pi r, where r = 9.
Thus, C = 27\pi. To find the angle AOB:
\theta = \frac{27}{9} \cdot 360° = 3 \cdot 360° = 1080°, but we convert it to radians: \theta ≈ 177°. Hence, the angle AOB that must be cut is 177°.