3.8 Numerical Methods (A-level only)

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Question 103

The concentration of a specific chemical pollutant in a reservoir, $P\,$ parts per million (ppm), is monitored over time. An environmental scientist models this concentration using the equation

60\left(\frac{t}{20}\right)^3 - 15\left(\frac{t}{20}\right)^4

where $t\,$ is the number of years after 1 January 2010.

ai.

The scientist predicts the pollutant will be completely dissipated ( $P = 0$ ) at time $T$ . Show that $T\,$ satisfies the equation

\sqrt[3]{80T^2 + \frac{128\,000}{T}}

[3]

aii.

Use the iterative formula $000Tn3\displaystyle T_{n+1} = \sqrt[3]{80T_n^2 + \frac{128\,000}{T_n}}$ , with $T_0 = 75$ , to find the values of $T_1, T_2,\,$ and $T_3\,$ to three decimal places.

[3]

aiii.

Explain the relevance of using $T_0 = 75$ in terms of the date.

[1]

On 1 January 2010, a natural neutralizing process in the reservoir begins. Its effectiveness at counteracting the pollutant is modeled by

\times 1.05^t

Use the models to show that the pollutant concentration $P\,$ and the neutralizing effectiveness $N\,$ will be equal during the year 2028.

[3]

3.8 Numerical Methods (A-level only)

Question 103

The concentration of a specific chemical pollutant in a reservoir, $P\,$ parts per million (ppm), is monitored over time. An environmental scientist models this concentration using the equation

60\left(\frac{t}{20}\right)^3 - 15\left(\frac{t}{20}\right)^4

where $t\,$ is the number of years after 1 January 2010.

ai.

The scientist predicts the pollutant will be completely dissipated ( $P = 0$ ) at time $T$ . Show that $T\,$ satisfies the equation

\sqrt[3]{80T^2 + \frac{128\,000}{T}}

[3]

aii.

Use the iterative formula $000Tn3\displaystyle T_{n+1} = \sqrt[3]{80T_n^2 + \frac{128\,000}{T_n}}$ , with $T_0 = 75$ , to find the values of $T_1, T_2,\,$ and $T_3\,$ to three decimal places.

[3]

aiii.

Explain the relevance of using $T_0 = 75$ in terms of the date.

[1]

On 1 January 2010, a natural neutralizing process in the reservoir begins. Its effectiveness at counteracting the pollutant is modeled by

\times 1.05^t

Use the models to show that the pollutant concentration $P\,$ and the neutralizing effectiveness $N\,$ will be equal during the year 2028.

[3]

3.8 Numerical Methods (A-level only)

Question 103

Markscheme

3.8 Numerical Methods (A-level only)

Question 103

Markscheme

3.8 Numerical Methods (A-level only)

Question 103

3.8 Numerical Methods (A-level only) Questions

3.8 Numerical Methods (A-level only)

Question 103

3.8 Numerical Methods (A-level only) Questions