IB Mathematics Applications & Interpretation Topic SL…

The severity of rockbursts in a mine is measured on a local magnitude scale, with values usually ranging from $0$ to $4$ , where $4$ represents the highest severity.

For one mining site, the expected annual number, $R$ , of rockbursts reaching at least magnitude $m$ is modeled by $\log_{10} R = p - 2m$ , where . At this site, there are on average rockbursts per year with a magnitude of or greater.

(a) the value of $p$ .

(b) the value of $d$ .

(e) the duration, to the nearest second, from the moment the mould was removed from the warming cabinet until the wax reaches $45^\circ\text{C}$ .

(f) the significance of the value 22 within this mathematical model.

The area covered by an invasive plant in a pond is modelled by the function, $A(w)=c\times 2^w$ , $w\ge 0$ where $c$ is a constant and $w$ represents the time in weeks since the beginning of the survey. It is observed that the area covered is 384 m $^2$ after 6 weeks.

A candle maker uses a hemispherical mould to form a decorative candle. The mould has radius 9 cm and is filled exactly with molten wax. Before melting, the same wax was shaped as a right circular cone with base radius 7 cm.

After the filled mould is removed from a warming cabinet, the temperature, $W$ , of the wax in degrees Celsius (°C), is modeled by the function:

$W(t)=a(1.28)^{-t}+22,\ t \geqslant 0$

where $a$ is a constant and $t$ is the time in minutes after removal. The wax is initially at $150^\circ\text{C}$ and can be removed from the mould once it cools to $45^\circ\text{C}$ .

(a) the volume of the mould.

(b) the height of the cone of wax, providing your answer in cm rounded to one decimal place.

(d) the temperature of the wax 5 minutes after it was removed from the warming cabinet.

The pH, $p$ , of a solution depends on its hydrogen ion concentration, $c$ , expressed in mol dm $^{-3}$ . This relationship is defined by the equation: $p=-\log_{10}(c)$ , for $c>0$ .

(a) Find the pH of a solution with hydrogen ion concentration $6.3 \times 10^{-5}$ mol dm $^{-3}$ .

A telescope measures the brightness of stars as seen from Earth using the apparent magnitude formula $m=-2.5\log_{10}(F/F_0)$ , where $F$ is the flux in watts per square metre and $F_0=3.6\times10^{-8}$ is a reference flux in the same units. Lower magnitude values correspond to brighter stars. One star in an image has flux $9.0\times10^{-11}$ watts per square metre. A second star in the same image has flux that is $25$ times that of the first star.

(a) Find the apparent magnitude of the first star.

(b) whether the second star is brighter or dimmer than the first star. Provide a mathematical justification for your conclusion.

(d) the expected time interval, rounded to the nearest year, between the occurrence of a magnitude $3.8$ rockburst and the next rockburst of at least that magnitude.

(a) the value of $c$ .

(b) the meaning of $c$ within this context.

(b) If a solution has a pH of $3.5$ , its hydrogen ion concentration $c$ .

IB Mathematics Applications & Interpretation SL 1.5—exponents and Logarithms Questionbank

IB Mathematics Applications & Interpretation SL 1.5—exponents and Logarithms Questionbank

IB Mathematics Applications & Interpretation SL 1.5—exponents and Logarithms Questionbank

IB Mathematics Applications & Interpretation SL 1.5—exponents and Logarithms Questionbank

IB Mathematics Applications & Interpretation (AI) Topic SL 1.5—exponents and Logarithms Questionbank