IB Mathematics Applications & Interpretation Topic…

An ideal amplifier multiplies a signal amplitude by its gain $G\gt0$ . For calculations, use $\log_{10}2\approx0.30103$ and $\log_{10}3\approx0.47712$ .

(a) Use the laws of logarithms to $\log_{10}G$ for gains $G=108$ and $G=\frac{8}{27}$ . Give both answers to four decimal places.

(b) A gain $G$ is set by $G=2^m3^n$ , where $m$ and $n$ are integers with . The measured value of , rounded to five decimal places, is . , and .

(c) The signal passes through $N$ identical amplifiers with the gain from part (b). the least positive integer $N$ for which the total gain exceeds 1080.

Lengths in this problem are measured in centimetres. Consider the function $r(x)=1+\sqrt{x}$ , defined for $0\leqslant x\leqslant 4$ . A candle is formed by rotating the region between the curve $y=r(x)$ and the $x$ -axis through $360^\circ$ about the $x$ -axis.

(a) the total volume of the candle.

(b) As the candle burns, the volume $W$ cm $^3$ of wax remaining after $t$ hours is modeled by $W=\frac{68\pi}{3}e^{-0.25t}$ . the time elapsed until half the candle remains.

A laboratory calibrates a sensor at a positive concentration ratio $x$ , measured relative to a reference concentration. At a sensitivity setting $k$ , zero calibration error occurs when

$\left(\log_{10}x\right)^2-k\log_{10}x+4=0.$

The model permits all $x\gt0$ and all real $k$ .

(a) Classify the values of $k$ for which the equation has zero, one or two positive solutions for $x$ .

(b) When there are two correctly calibrated concentration ratios $x_1$ and $x_2$ , show that $x_1x_2=10^k$ .

(c) Suppose that $k\gt 0$ and the two solutions satisfy $\dfrac{x_2}{x_1}=100$ , where $x_2\gt x_1$ . $k$ , $x_1$ and $x_2$ exactly.

A flare's height is modelled by $h(t)=-t^3+7t^2+2t+1$ , where $h$ is in metres and $t$ is in seconds. In the first two parts, use technology where appropriate.

(a) Find the time when the flare hits the ground, giving your answer correct to three significant figures.

(b) A signal reading is modelled by $P(u)=u^3-4u^2-11u+20$ . Write down an equation whose solutions are the $u$ -values for which the signal reading is $8$ , and solve it.

(c) The cost of a subscription is modelled by $C(t)=240(1.09)^t$ , where $C$ is in dollars and $t$ is in years. Find the value of $t$ when the cost reaches $600$ .

Consider the function $h(x)=\ln m-\ln((x-1)(5-x))$ , defined on the interval $1<x<5$ , where $m>0$ .

It is known that the equation $h(x)=0$ has exactly one unique solution.

(d) Hence the first whole year for which the cost exceeds $600$ .

IB Mathematics Applications & Interpretation AHL 1.9—log Laws Questionbank

IB Mathematics Applications & Interpretation AHL 1.9—log Laws Questionbank

IB Mathematics Applications & Interpretation AHL 1.9—log Laws Questionbank

IB Mathematics Applications & Interpretation AHL 1.9—log Laws Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 1.9—log Laws Questionbank