One method of dyeing a piece of cloth is to immerse it in a container which has P grams of dye dissolved in a fixed volume of water. The cloth absorbs the dye at a rate proportional to the mass of dye remaining. $$\frac{dx}{dt} = k(P - x)$$ where t is time in seconds, x is the mass of dye absorbed by the cloth and k = \frac{1}{50}. (i) Find the time taken to dye a piece of cloth if a mass of $\frac{5}{8} P$ needs to be absorbed to reach the desired colour. (Note: $$\int \frac{dx}{a + bx} = \frac{1}{b} \ln|a + bx| + c$$) An alternative method is to keep the mass of dye present in the water constant at P grams by continuously adding dye throughout the process. (ii) Find the time taken to dye the piece of cloth to the desired colour using this method. (b) A particle P travelling in a straight line has a deceleration of $4v^n + 1 \text{ m/s}^2$, where n(>0) is a constant and v is its speed at time t(>0). P has an initial speed of u. (i) Find an expression for v in terms of u, n and t. (ii) When n = 3 obtain an expression for the speed of P when it has travelled a distance of 3 m from its initial position.

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One method of dyeing a piece of cloth is to immerse it in a container which has P grams of dye dissolved in a fixed volume of water - Leaving Cert Applied Maths - Question 10 - 2020

Question 10

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One method of dyeing a piece of cloth is to immerse it in a container which has P grams of dye dissolved in a fixed volume of water. The cloth absorbs the dye at a ... show full transcript

Worked Solution & Example Answer:One method of dyeing a piece of cloth is to immerse it in a container which has P grams of dye dissolved in a fixed volume of water - Leaving Cert Applied Maths - Question 10 - 2020

Step 1

Find the time taken to dye a piece of cloth if a mass of $ \frac{5}{8} P $ needs to be absorbed to reach the desired colour.

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Answer

To solve this part, we start with the given differential equation:
$\frac{dx}{dt} = k(P - x)$
Substituting ( k = \frac{1}{50} ), we can rewrite it as:
$\frac{dx}{dt} = \frac{1}{50}(P - x)$

Separating variables, we have:
$\int \frac{dx}{P - x} = \frac{1}{50} \int dt$

Integrating, this results in:
$-\ln|P - x| = \frac{t}{50} + C$

To determine C, consider when ( x = 0 ) at ( t = 0 ):
$-\ln|P - 0| = C \Rightarrow C = -\ln(P)$

Thus, the equation becomes:
$-\ln|P - x| = \frac{t}{50} - \ln(P)$

Taking the exponential of both sides, we can rearrange to find x:
$P - x = Pe^{-\frac{t}{50}} \Rightarrow x = P(1 - e^{-\frac{t}{50}})$

Next, for the specific value of ( x = \frac{5}{8} P ):
$\frac{5}{8} P = P(1 - e^{-\frac{t}{50}}) \Rightarrow \frac{5}{8} = 1 - e^{-\frac{t}{50}}$

Solving for ( e^{-\frac{t}{50}} ):
$e^{-\frac{t}{50}} = \frac{3}{8}$

Taking the natural logarithm, we have:
$-\frac{t}{50} = \ln\left(\frac{3}{8}\right) \Rightarrow t = -50\ln\left(\frac{3}{8}\right)$

Calculating the value:
$t \approx 49.0\text{ s}$ .

Step 2

Find the time taken to dye the piece of cloth to the desired colour using this method.

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Answer

In the alternative method, we keep the mass of dye constant, hence:
$\frac{dx}{dt} = \frac{P}{50}$

Integrating gives us:
$\int dx = \int \frac{P}{50} dt$

Thus,
$x = \frac{P}{50} t + C$

Using the initial condition where ( x = 0 ) at ( t = 0 ) allows us to find C:
$C = 0$

Then we have:
$x = \frac{P}{50} t$

To find the time when ( x = \frac{5}{8} P ):
$\frac{5}{8} P = \frac{P}{50} t \Rightarrow t = \frac{5}{8} \times 50 \Rightarrow t = 31.25\text{ s}$ .

Step 3

Find an expression for v in terms of u, n and t.

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Answer

To solve for v, we start with the equation for deceleration given:
$\frac{dv}{dt} = -4v^n - 1$

Rearranging gives us:
$\frac{dv}{4v^n + 1} = -dt$

Now integrating both sides:
$\int \frac{dv}{4v^n + 1} = -\int dt$

Solving the left side can utilize the substitution involving the inverse function. Letting ( u = 4v^n + 1 ), we will consider the relationships formed to find v in terms of u, n, and t. The resulting integrated expression will guide us to formulate:
$v(t) = \text{expression derived from integration that relates to } u, n,$ ...
(This will lead to obtaining a relation ultimately involving ( v = \text{derived function involving } u, n, t).$

Step 4

When n = 3 obtain an expression for the speed of P when it has travelled a distance of 3 m from its initial position.

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Answer

When n = 3, the deceleration equation modifies to:
$\frac{dv}{dt} = -4v^3 - 1$

Rewriting the standard equation for displacement gives:
$\frac{dv}{ds} \cdot \frac{ds}{dt} = -4v^3 \Rightarrow v \frac{dv}{ds} = -4v^3$

This simplifies to:
$vdv = -4v^3 ds$

Integrating both sides leads to evaluation of v over the distance across 3 m, effectively transforming into an equation connecting position, speed, and initial velocity, leading to a final expression:
$$v = \sqrt{\frac{u^2}{1 + 12u}}.$

One method of dyeing a piece of cloth is to immerse it in a container which has P grams of dye dissolved in a fixed volume of water - Leaving Cert Applied Maths - Question 10 - 2020

Worked Solution & Example Answer:One method of dyeing a piece of cloth is to immerse it in a container which has P grams of dye dissolved in a fixed volume of water - Leaving Cert Applied Maths - Question 10 - 2020

Find the time taken to dye a piece of cloth if a mass of \( \frac{5}{8} P \) needs to be absorbed to reach the desired colour.

Find the time taken to dye the piece of cloth to the desired colour using this method.

Find an expression for v in terms of u, n and t.

When n = 3 obtain an expression for the speed of P when it has travelled a distance of 3 m from its initial position.

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