IB Maths AI HL AI 1.3 Exponents, logarithms and equations questions

A laboratory compares the hydrogen ion concentrations of two aqueous solutions. Figure 1 shows the measured concentrations. For these solutions, the one with the greater hydrogen ion concentration is more acidic.

For the pH calculations, use

$$\mathrm{pH}=-\log_{10}(C),$$

where $C$ is the numerical value of the hydrogen ion concentration in $\mathrm{mol\,dm^{-3}}$

A theatre records the total revenue from three group ticket sales. Let $a$ $s$ and $c$ be the prices, in dollars, of one adult, student and child ticket, respectively. The records are shown in Figure 1.

For a later performance, the regular adult ticket price is $p$ dollars. If the discount is $d$ dollars, the estimated number of discounted adult tickets sold is $4d+12$ The revenue from these tickets is modelled by

$$R=(p-d)(4d+12),$$

where $0<d<p$

A fountain jet is modelled, for $0\leq x\leq 6$ by

$$h=ax^2+bx+c,$$

where $x$ is the horizontal distance from the nozzle, in metres, and $h$ is the height above the ground, in metres. The model passes through the points shown.

During an acoustic test, sound level $L$ in decibels (dB), is defined by

$$L=10\log_{10}\left(\frac{I}{I_0}\right),$$

where $I$ is the sound intensity and $I_0$ is a fixed reference intensity. When machines operate together, their sound intensities add.

Machine A alone produces a sound level of $67.4$ dB and machine B alone produces a sound level of $72.1$ dB.