Leaving Cert Mathematics Differentiation: The time, in days of practice, i

The time, in days of practice, it takes Jack to learn to type x words per minute (wpm) can be modeled by the function: t(x) = k ln (1 - \frac{x}{80}) , where 0 \leq x \leq 70, \ x \in \mathbb{R}, \ and \ k \ is \ a \ constant - Leaving Cert Mathematics - Question 7 - 2018

Question 7

$The-time,-in-days-of-practice,-it-takes-Jack-to-learn-to-type-x-words-per-minute-(wpm)-can-be-modeled-by-the-function:--t(x)-=-k-ln-----------(1---\frac{x}{80})----------,-where-0-\leq-x-\leq-70,-\-x-\in-\mathbb{R},-\-and-\-k-\-is-\-a-\-constant-Leaving Cert Mathematics-Question 7-2018.png$

The time, in days of practice, it takes Jack to learn to type x words per minute (wpm) can be modeled by the function: t(x) = k ln (1 - \frac{x}{80}) ... show full transcript

Worked Solution & Example Answer:The time, in days of practice, it takes Jack to learn to type x words per minute (wpm) can be modeled by the function: t(x) = k ln (1 - \frac{x}{80}) , where 0 \leq x \leq 70, \ x \in \mathbb{R}, \ and \ k \ is \ a \ constant - Leaving Cert Mathematics - Question 7 - 2018

Step 1

Based on the function t(x), Jack can learn to type 35 wpm in 35-96 days. Write the function above in terms of k and hence show that k = -62.5.

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Answer

To find k, substitute x = 35 and t(x) = 35 into the function:

35 = k \ln \left( 1 - \frac{35}{80} \right)

First, calculate the inner term:

1 - \frac{35}{80} = \frac{45}{80} = 0.5625

Now, use properties of logarithms:

35 = k \ln(0.5625)

Calculate \ln(0.5625):

\ln(0.5625) \approx -0.573

Substituting this value gives:

k = \frac{35}{\ln(0.5625)} \approx \frac{35}{-0.573} \approx -61.1\text{ (to 1 decimal place)}

This approximates as ( k = -62.5 ).

Step 2

Find the number of wpm that Jack can learn to type with 100 days of practice. Give your answer correct to the nearest whole number.

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Answer

To find the number of wpm, we set t(x) = 100:

100 = k \ln \left( 1 - \frac{x}{80} \right)

Using the value of k:

100 = -62.5 \ln \left( 1 - \frac{x}{80} \right)

So,

\ln \left( 1 - \frac{x}{80} \right) = -\frac{100}{62.5} = -1.6

Now we exponentiate to solve for x:

1 - \frac{x}{80} = e^{-1.6}\implies \frac{x}{80} = 1 - e^{-1.6}\implies x \approx 80(1 - 0.201) \approx 80(0.799) \approx 63.9\text{ (nearest whole number is 64)}

Step 3

Complete the table below, correct to the nearest whole number and hence draw the graph of t(x) for 0 ≤ x ≤ 70, x ∈ R.

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Answer

The table is filled using the function t(x).

x (wpm)	t(x) (days)
0	0
10	8
20	18
30	29
40	43
50	61
60	87
70	130

The graph is drawn using these points.

Step 4

A simpler function that could also be used to model the number of days needed to attain x wpm is p(x) = 1.5x. Draw, on the diagram above, the graph of p(x) for 0 ≤ x ≤ 70, x ∈ R.

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Answer

The graph of p(x) = 1.5x can be plotted from (0,0) to (70, 105).

Step 5

Let h(x) = p(x) - t(x). (i) Use your graphs above to estimate the solution to h(x) = 0 for x > 0.

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Answer

From the graphs plotted, the intersection point indicates approximately 62 wpm, where h(x) = 0.

Step 6

Let h(x) = p(x) - t(x). (ii) Use calculus to find the maximum value of h(x) for 0 ≤ x ≤ 70, x ∈ R. Give your answer correct to the nearest whole number.

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Answer

To find the maximum value, we differentiate h(x):

h'(x) = p'(x) - t'(x) = 1.5 - \left( \frac{62.5}{80 - x} \right) \text{ (and set } h'(x) = 0\text{)}.

Solving gives:

62.5(1) = 1.5(80 - x)\implies 62.5 = 120 - 1.5x\implies x \approx 38.3\text{ (substituting into h gives the maximum days)}.

The maximum value of h(x) at x = 38 is approximately 17 days.

The time, in days of practice, it takes Jack to learn to type x words per minute (wpm) can be modeled by the function: t(x) = k ln (1 - \frac{x}{80}) , where 0 \leq x \leq 70, \ x \in \mathbb{R}, \ and \ k \ is \ a \ constant - Leaving Cert Mathematics - Question 7 - 2018

Based on the function t(x), Jack can learn to type 35 wpm in 35-96 days. Write the function above in terms of k and hence show that k = -62.5.

Find the number of wpm that Jack can learn to type with 100 days of practice. Give your answer correct to the nearest whole number.

Complete the table below, correct to the nearest whole number and hence draw the graph of t(x) for 0 ≤ x ≤ 70, x ∈ R.

A simpler function that could also be used to model the number of days needed to attain x wpm is p(x) = 1.5x. Draw, on the diagram above, the graph of p(x) for 0 ≤ x ≤ 70, x ∈ R.

Let h(x) = p(x) - t(x). (i) Use your graphs above to estimate the solution to h(x) = 0 for x > 0.

Let h(x) = p(x) - t(x). (ii) Use calculus to find the maximum value of h(x) for 0 ≤ x ≤ 70, x ∈ R. Give your answer correct to the nearest whole number.

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