5. (a) Find the exact value of the volume of the solid of revolution formed when the region bounded by the curve $y = ext{sin}(2x)$, the x-axis and the line $x = \frac{\pi}{8}$ is rotated about the x-axis. (b) Two chords of a circle, AB and CD, intersect at E. The perpendiculars to AB at A and CD at D intersect at P. The line PE meets BC at Q, as shown in the diagram. (i) Explain why DPAE is a cyclic quadrilateral. (ii) Prove that $\angle APE = \angle ABC.$ (iii) Deduce that PQ is perpendicular to BC. (c) A particle moves in a straight line and its position at time t is given by $$x = 5 + \sqrt{3}\text{sin}(3t) - \text{cos}(3t).$$ (i) Express $\sqrt{3}\text{sin}(3t)$ in the form R\text{sin}(3t - \alpha), where \alpha is in radians. (ii) The particle assumes simple harmonic motion. Find the amplitude and the circumference of the motion. (iii) When does the particle first reach its maximum speed although some were using vague (i) and (ii), it became clear that misunderstandings often occur when given $x(0) = 5$ and finding a different answer in part (iii). A common error was to state that maximum speed occurs when $v = 0.$

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5. (a) Find the exact value of the volume of the solid of revolution formed when the region bounded by the curve $y = ext{sin}(2x)$, the x-axis and the line $x = \frac{\pi}{8}$ is rotated about the x-axis - HSC - SSCE Mathematics Extension 1 - Question 5 - 2005 - Paper 1

Question 5

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5. (a) Find the exact value of the volume of the solid of revolution formed when the region bounded by the curve $y = ext{sin}(2x)$, the x-axis and the line $x = \f... show full transcript

Worked Solution & Example Answer:5. (a) Find the exact value of the volume of the solid of revolution formed when the region bounded by the curve $y = ext{sin}(2x)$, the x-axis and the line $x = \frac{\pi}{8}$ is rotated about the x-axis - HSC - SSCE Mathematics Extension 1 - Question 5 - 2005 - Paper 1

Step 1

Find the exact value of the volume of the solid of revolution formed when the region bounded by the curve $y = ext{sin}(2x)$, the x-axis and the line $x = \frac{\pi}{8}$ is rotated about the x-axis.

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Answer

To find the volume of the solid of revolution, we can use the formula for the volume of revolution around the x-axis:

$V = \pi \int_{a}^{b} f(x)^2 \, dx$

In this case, we have:

$f(x) = \text{sin}(2x)$
$a = 0$
$b = \frac{\pi}{8}$ .

Thus, the volume becomes:

$V = \pi \int_{0}^{\frac{\pi}{8}} \text{sin}^2(2x) \, dx$

Using the identity $\text{sin}^2(x) = \frac{1 - \text{cos}(2x)}{2}$ , we can rewrite $\text{sin}^2(2x)$ as:

$\text{sin}^2(2x) = \frac{1 - \text{cos}(4x)}{2}$

Then, substituting this back into the volume equation, we have:

$V = \pi \int_{0}^{\frac{\pi}{8}} \frac{1 - \text{cos}(4x)}{2} \, dx$

Now, calculate the integral:

$= \frac{\pi}{2} \int_{0}^{\frac{\pi}{8}} (1 - \text{cos}(4x)) \, dx$

This evaluates to:

$= \frac{\pi}{2} \left[ x - \frac{\text{sin}(4x)}{4} \right]_{0}^{\frac{\pi}{8}}$

Substituting in the limits gives:

$= \frac{\pi}{2} \left( \frac{\pi}{8} - 0 \right) = \frac{\pi^2}{16}$

So, the volume of the solid of revolution is:

$V = \frac{\pi^2}{16}.$

Step 2

Explain why DPAE is a cyclic quadrilateral.

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Answer

A cyclic quadrilateral is one where all vertices lie on the circumference of a circle. In quadrilateral DPAE, the opposite angles add up to 180 degrees. Specifically, we have:

$\angle DAP + \angle DEP = 180^\circ$
$\angle DPE + \angle AEP = 180^\circ$ As such, DPAE satisfies the property of cyclic quadrilaterals.

Step 3

Prove that $\angle APE = \angle ABC.$

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Answer

In triangle ABE, angle ABE subtends the chord AE at points A and E. By the property of angles subtended by the same chord, we have: $\angle ABE = \angle APE$ In triangle ABC, angle ABC also subtends the same chord BC at points B and C. Therefore, we can conclude that: $\angle APE = \angle ABC.$

Step 4

Deduce that PQ is perpendicular to BC.

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Answer

Since $\angle APE = \angle ABC$ and we know that both angles are equal, triangles APE and ABC are similar by AA criterion. Thus, if angle APE is equal to angle ABC and given that line segments ARE both perpendicular (due to the properties of cyclic quadrilaterals), we can deduce that PQ must be perpendicular to BC.

Step 5

Express $\sqrt{3}\text{sin}(3t)$ in the form R\text{sin}(3t - \alpha), where \alpha is in radians.

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Answer

To convert $\sqrt{3}\text{sin}(3t)$ into the form R\text{sin}(3t - \alpha), we identify R and \alpha such that: $R\text{sin}(3t - \alpha) = R(\text{sin}(3t)\text{cos}(\alpha) - \text{cos}(3t)\text{sin}(\alpha))$ Comparing coefficients:

$R\text{cos}(\alpha) = \sqrt{3}$
$R\text{sin}(\alpha) = 1$ To find R, use: $R = \sqrt{(\sqrt{3})^2 + (1)^2} = 2$ Now to find \alpha: $\tan(\alpha) = \frac{1}{\sqrt{3}} \\ \Rightarrow \alpha = \frac{\pi}{6} \\$ Thus, the expression becomes: $2\text{sin}(3t - \frac{\pi}{6}).$

Step 6

The particle assumes simple harmonic motion. Find the amplitude and the circumference of the motion.

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Answer

For a particle exhibiting simple harmonic motion described by: $x(t) = 5 + \sqrt{3}\text{sin}(3t) - \text{cos}(3t),$ The amplitude is given by the maximum value of the terms that multiply the sinusoidal functions. Hence, the amplitude can be calculated as: $A = \sqrt{(\sqrt{3})^2 + (1)^2} = 2.$ The circumference of the motion is given by the formula: $C = 2\pi A = 2\pi (2) = 4\pi.$

Step 7

When does the particle first reach its maximum speed?

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Answer

The speed of the particle is given by the derivative of position: $v(t) = \frac{dx}{dt}$ For maximum speed, set the derivative of speed equal to zero while observing max speed times: $v(t) = 3\sqrt{3}\text{cos}(3t) + 3\text{sin}(3t) = 0.$ Thus, when $3t$ is such that: $3t = k\pi + \frac{\pi}{2}, k \in \mathbb{Z}$ From this, solving for $t$ yields: $t = \frac{k\pi + \frac{\pi}{2}}{3}.$ Thus the first occurrence happens at: $t = \frac{\pi}{6}$ . This corresponds to the conditions set in parts (i) and (ii).

5. (a) Find the exact value of the volume of the solid of revolution formed when the region bounded by the curve $y = ext{sin}(2x)$, the x-axis and the line $x = \frac{\pi}{8}$ is rotated about the x-axis - HSC - SSCE Mathematics Extension 1 - Question 5 - 2005 - Paper 1

Find the exact value of the volume of the solid of revolution formed when the region bounded by the curve $y = ext{sin}(2x)$, the x-axis and the line $x = \frac{\pi}{8}$ is rotated about the x-axis.

Explain why DPAE is a cyclic quadrilateral.

Prove that $\angle APE = \angle ABC.$

Deduce that PQ is perpendicular to BC.

Express $\sqrt{3}\text{sin}(3t)$ in the form R\text{sin}(3t - \alpha), where \alpha is in radians.

The particle assumes simple harmonic motion. Find the amplitude and the circumference of the motion.

When does the particle first reach its maximum speed?

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