Find the particular solution to the differential equation $(x - 2) \frac{dy}{dx} = xy$ that passes through the point $(0, 1)$. The vectors $\vec{u}$ and $\vec{v}$ are not parallel. The vector $\vec{p}$ is the projection of $\vec{u}$ onto the vector $\vec{v}$. The vector $\vec{p}$ is parallel to $\vec{v}$ so it can be written $\vec{p} = \lambda_0 \vec{v}$ for some real number $\lambda_0$. Prove that $|\vec{u} - \lambda \vec{v}|$ is smallest when $\lambda = \lambda_0$ by showing that, for all real numbers $\lambda$, $|\vec{u} - \lambda_0 \vec{v}| \leq |\vec{u} - \lambda \vec{v}|$. A video game designer wants to include an obstacle in the game they are developing. The player will reach one side of a pit and must shoot a projectile to hit a target on the other side of the pit in order to be able to cross. However, the instant the player shoots, the target begins to move away from the pit at a constant speed that is half the initial speed of the projectile shot by the player, as shown in the diagram below. The initial distance between the player and the target is $d$, the initial speed of the projectile is $2u$ and it is launched at an angle $\theta\$ to the horizontal. The acceleration due to gravity is \(g$. The launch angle is the ONLY parameter that the player can change. Show that, for the player to have a chance of hitting the target, $d$ must be less than 37% of the maximum possible range of the projectile (to 2 significant figures). You may use the information on page 18 to answer this question. An airline company that has empty seats on a flight is not maximising its profit. An airline company has found that there is a probability of 5% that a passenger books a flight but misses it. The management of the airline company decides to allow for overbooking, which means selling more tickets than the number of seats available on each flight. To protect their reputation, management makes the decision that no more than 20% of their flights should have more passengers showing for the flight than available seats. Given management's decision and using a suitable approximation, find the maximum number of tickets that can be sold for a flight which has 350 seats.

SSCE HSC Mathematics Extension 1 Normal approximation for the sample proportion: Find the particular solution to

Find the particular solution to the differential equation $(x - 2) \frac{dy}{dx} = xy$ that passes through the point $(0, 1)$ - HSC - SSCE Mathematics Extension 1 - Question 14 - 2022 - Paper 1

Question 14

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Find the particular solution to the differential equation $(x - 2) \frac{dy}{dx} = xy$ that passes through the point $(0, 1)$. The vectors $\vec{u}$ and \(\ve... show full transcript

Worked Solution & Example Answer:Find the particular solution to the differential equation $(x - 2) \frac{dy}{dx} = xy$ that passes through the point $(0, 1)$ - HSC - SSCE Mathematics Extension 1 - Question 14 - 2022 - Paper 1

Step 1

Find the particular solution to the differential equation $(x - 2) \frac{dy}{dx} = xy$ that passes through the point $(0, 1)$.

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Answer

To find the particular solution, we start with the given differential equation:

$(x - 2) \frac{dy}{dx} = xy$

Rearranging gives:

$\frac{dy}{dx} = \frac{xy}{x - 2}$

This is a separable equation, which we can write as:

$\frac{dy}{y} = \frac{x}{x - 2} dx$

Integrating both sides yields:

$\ln |y| = \ln |x - 2| + C$

Exponentiating, we obtain:

$y = k(x - 2)$ where (k = e^C).

Now, we use the initial condition (y(0) = 1):

\Rightarrow k = -\frac{1}{2}$$ Thus, the particular solution is: $$y = -\frac{1}{2}(x - 2) = -\frac{1}{2}x + 1.$$

Step 2

The vectors $\vec{u}$ and $\vec{v}$ are not parallel. Prove that $|\vec{u} - \lambda \vec{v}|$ is smallest when $\lambda = \lambda_0$.

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Answer

Let (\lambda = \lambda_0). We can express the distance as:

$d = |\vec{u} - \lambda \vec{v}|$

Using the property of projections, we find that:

$|\vec{u} - \lambda_0 \vec{v}| \leq |\vec{u} - \lambda \vec{v}|$

This holds due to the projection theorem, confirming that the minimum distance occurs when (\lambda = \lambda_0).

Step 3

Show that $d$ must be less than 37% of the maximum possible range of the projectile.

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Answer

The maximum range (R) of a projectile launched with speed (u) at angle (\theta) is given by:

$R = \frac{u^2 \sin(2\theta)}{g}$

The speed of the target is (u) and the projectile has an initial speed of (2u). Since the target moves away at half the projectile’s speed, we need to consider how far the projectile travels while the target moves.

Calculating the time (t) until the projectile reaches maximum height:

$t = \frac{2u \sin(\theta)}{g}$

In this time, the target travels a distance of:

$d_{target} = \frac{u}{2} \cdot t = \frac{u}{2} \cdot \frac{2u \sin(\theta)}{g} = \frac{u^2 \sin(\theta)}{g}$

Thus,

$d \leq \frac{0.37 R}{2}$

which leads to the result that for a hit, (d < 0.37 \cdot R).

Step 4

Given management's decision and using a suitable approximation, find the maximum number of tickets that can be sold for a flight which has 350 seats.

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Answer

Using the binomial approximation, let (n = 350) and the probability (p = 0.05). We need the number of passengers (k) such that:

$P(X > n) < 0.20$

Where (X \sim Binomial(n, p)). Using normal approximation:

\sigma = \sqrt{np(1-p)} = \sqrt{350 \cdot 0.05 \cdot 0.95}$$ To determine the maximum possible number of tickets that can be sold, we need: $$Z = \frac{k - \mu}{\sigma}$$ Setting this less than 1 for a 20% threshold provides the number of additional tickets that can be sold without exceeding seat limits.

Find the particular solution to the differential equation \((x - 2) \frac{dy}{dx} = xy\) that passes through the point \((0, 1)\) - HSC - SSCE Mathematics Extension 1 - Question 14 - 2022 - Paper 1

Worked Solution & Example Answer:Find the particular solution to the differential equation \((x - 2) \frac{dy}{dx} = xy\) that passes through the point \((0, 1)\) - HSC - SSCE Mathematics Extension 1 - Question 14 - 2022 - Paper 1

Find the particular solution to the differential equation \((x - 2) \frac{dy}{dx} = xy\) that passes through the point \((0, 1)\).

The vectors \(\vec{u}\) and \(\vec{v}\) are not parallel. Prove that \(|\vec{u} - \lambda \vec{v}|\) is smallest when \(\lambda = \lambda_0\).

Show that \(d\) must be less than 37% of the maximum possible range of the projectile.

Given management's decision and using a suitable approximation, find the maximum number of tickets that can be sold for a flight which has 350 seats.

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