IB DP Math MAI HL : IB Style Mock Exams – Set 7 Paper 1

Question

[Maximum mark: 4]
Cylindrical logs of length $2.2 \mathrm{~m}$ and radius $7.6 \mathrm{~cm}$ are used to construct a fence line. Part of the logs are removed, as illustrated in the diagram below.

Find the volume of each log.

Answer/Explanation

First we find the area of the sector remaining. Using the area of a sector formula, we get
$
\begin{aligned}
A & =\frac{\theta}{360} \pi r^2 \\
& =\frac{360-60}{360} \pi\left(7.6^2\right) \\
& =\frac{300}{360} \pi\left(7.6^2\right) \\
& \approx 151.2 \mathrm{~cm}^2
\end{aligned}
$

The volume of the $\log$ can now be found by
$
\begin{aligned}
V & =A l \\
& =151.2(220) \\
& \approx 33300 \mathrm{~cm}^3
\end{aligned}
$

Question

[Maximum mark: 6]
$\text{In this question give all answers correct to two decimal places.}$
Charlie deposits 8000 Canadian dollars (CAD) into a bank account. The bank pays a nominal annual interest rate of $5 \%$, compounded semi-monthly.

(a) Find the amount of interest that Charlie will earn over the next 2 years.                            [3]

Oscar also deposits CAD into a bank account. His bank pays a nominal annual interest rate of $6 \%$, compounded quarterly. In 2 years, the total amount in Oscar’s account will be $\$ 8000$ CAD.

(b) Find the amount that Oscar deposits in the bank account.                         [3]

Answer/Explanation

(a) We have
$
\begin{array}{|c|c|c|c|c|}
\hline \mathrm{PV} & r & k & n & \mathrm{FV} \\
\hline 8000 \mathrm{CAD} & 5 \% & 24 & 2 & \\
\hline
\end{array}
$

Hence, using the compound interest formula, we get
$
\begin{aligned}
\mathrm{FV} & =\mathrm{PV}\left(1+\frac{r}{100 k}\right)^{k n} \\
& =8000\left(1+\frac{5}{100 \times 24}\right)^{24 \times 2} \\
& \approx 8840.45 \mathrm{CAD}
\end{aligned}
$

Hence the amount of interest earned is
$
\begin{aligned}
\mathrm{FV}-\mathrm{PV} & \approx 8840.45-8000 \\
& \approx 840.45 \mathrm{CAD}
\end{aligned}
$

(b) We have
$
\begin{array}{|c|c|c|c|c|}
\hline \mathrm{PV} & r & k & n & \mathrm{FV} \\
\hline & 6 \% & 4 & 2 & 8000 \mathrm{CAD} \\
\hline
\end{array}
$

Hence, using the compound interest formula, we obtain
$
\begin{aligned}
& 8000=\mathrm{PV}\left(1+\frac{6}{100 \times 4}\right)^{4 \times 2} \\
& \mathrm{PV} \approx 7101.69 \mathrm{CAD} \quad[\text { by using G.D.C.] }
\end{aligned}
$

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