1.10 Rational exponents questions for IB Maths AI

${125}^{\frac{2}{3}}$

${81}^{-\frac{3}{4}}$

${\left(\frac{16}{625}\right)}^{-\frac{1}{4}}$

Write down the volume of the balloon after 20 seconds.

Find the radius of the balloon after 20 seconds.

Show that $(r\circ V)(t)={\left({\displaystyle \frac{45t}{\pi }}\right)}^{\frac{1}{3}}$.

The surface area of the balloon after $t$ seconds is $(S\circ r\circ V)(t)$. Show that this can be written as $a{t}^{\frac{2}{3}}$ and find the value of $a$.

Find an expression for $(r\circ V{)}^{-1}(x)$ and state what it represents in context.

The balloon bursts when its radius reaches 15 cm. Find the time at which the balloon bursts.

Dr Okafor wants to plot the data. Suggest why she should use a logarithmic scale for the mass.

Find the values of ${\log }_{10}M$ and ${\log }_{10}H$ for the horse.

Let $X={\log }_{10}M$ and $Y={\log }_{10}H$.

Find Pearson's product-moment correlation coefficient, $r$, for $X$ and $Y$. Hence comment on whether a model of the form $H=k{M}^b$ is suitable.

Find the equation of the regression line of $Y$ on $X$.

Use your answer to part (d) to show that $H=k{M}^b$, and find the value of $k$ and the value of $b$.

A giraffe has a mass of 800 kg and a measured resting heart rate of 48 beats per minute.

Use your model to estimate the resting heart rate of the giraffe, and find the percentage error in this estimate.

Find the percentage decrease in resting heart rate that this model predicts when body mass is doubled.

Find the mass of a mammal with a predicted resting heart rate of 150 beats per minute.

Dr Okafor models the mean lifespan, $L$ years, of a mammal by $L=8M^{0.2}$. She investigates the total number of heartbeats, $N$, in a mammal's lifetime, assuming that the heart beats at its resting rate throughout. A year has 525 600 minutes.

Show that $N\approx 1.02\times {10}^9\times M^{-0.058}$.

Find the value of $N$ for a mouse and for an elephant. Give your answers in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

A popular science article claims that "every mammal gets about one billion heartbeats in its lifetime". Comment on this claim, with reference to your answers to parts (i)(i) and (i)(ii).

A blue whale has a mass of about 150 000 kg. State one reason why Dr Okafor's model may not give a reliable estimate of its heart rate.

Simplify $\frac{x^{\frac{2}{3}}\times x^{\frac{5}{6}}}{x^{-\frac{1}{2}}}$.

Write $\frac{6}{\sqrt[3]{x^4}}$ in the form $a{x}^n$, where $a,n\in \mathbb{Q}$.

Write $(x+5)\sqrt{x}$ as a sum of terms of the form $a{x}^n$.

Show that ${\log }_{10}V={\log }_{10}k+p{\log }_{10}D+q{\log }_{10}H$.

Use the data for Tree 1 to write down a linear equation in ${\log }_{10}k$, $p$ and $q$.

Find the values of $p$, $q$ and $k$.

A fourth tree has diameter 28 cm and height 18 m. Use the model to estimate the volume of wood in this tree.

When the fourth tree is felled, its volume is found to be $0.49{\text{ m}}^3$. Find the percentage error in the estimate from part (d).

State one reason why the model may give a poor estimate for some trees.

$x^{\frac{3}{2}}=125$, $x>0$

$9^x={27}^{x-1}$

$x^{-\frac{2}{3}}=\frac{1}{16}$, $x>0$