Simplifying expressions (with rational exponents): notes and practice questions

A landscape architect is designing a modular planter box for a new urban garden. The planter box has a base and top that are identical sectors of a circle, each with radius $r$ cm and angle $\theta$ radians. The height of the planter box is $h = 2$ cm. The total length of metal frame used for all edges of the planter box is $L = 20$ cm. This includes two circular arcs, four radial edges for the top and bottom sectors, and three vertical connecting edges.

(a) Show that $r = \frac{7}{2+\theta}$.

(b) The planter box is designed to hold soil, enclosing a volume, $V$.

(i) Find an expression for $V$ in terms of $\theta$.

(ii) Find the expression for $\frac{dV}{d\theta}$.

(iii) Solve algebraically $\frac{dV}{d\theta} = 0$ to find the value of $\theta$ that will maximize the volume, $V$.

A mathematical model for a physical phenomenon involves the expression $\left(\frac{1}{u} + 2\right)^2$.

(a) Expand $\left(\frac{1}{u} + 2\right)^2$.

The rate of change of a certain quantity is given by $f'(x) = \frac{1}{(x+1)^2} + \frac{4}{x+1} + 4$.

(b) Find the indefinite integral $\int \left( \frac{1}{(x+1)^2} + \frac{4}{x+1} + 4 \right) dx$.

A designer is creating a custom-shaped container. The cross-sectional profile of the container is defined by the curve $y = \frac{1}{x+1} + 2$ for $1 \le x \le 3$. The container is formed by rotating this region $2\pi$ about the x-axis.

(c) Calculate the volume of the solid formed. Give your answer in the form $\frac{\pi}{4}(a + b \ln(c) )$, where $a, b, c \in \mathbb{Z}$.

(a) A medical isotope used in diagnostic imaging decays over time. The time $t$ (in hours) since the isotope was prepared can be modelled by the equation $t = \frac{-12}{\ln 2} \ln(P)$, where $P$ is the proportion of the isotope remaining.

(i) Calculate the time $t$ (in hours) when the proportion of the isotope remaining is $P = 0.5$.

(ii) Calculate the time $t$ (in hours) when the proportion of the isotope remaining is $P = 0.25$.

(b) The isotope is considered ineffective if the proportion remaining drops below $0.05$. Find the time (in hours) when the isotope just becomes ineffective, giving your answer to one decimal place.

(c) Express $P$ in terms of $t$.

(d) Using your answer from part (c), find the proportion of the isotope remaining after $24$ hours. Give your answer to three significant figures.

(a) A scientist is modeling the growth rate of a specific enzyme in a culture. The instantaneous rate of change is given by the expression:

$\sqrt{12x^5} \times 5x^0 \times 2x^3$

Simplify this expression, assuming $x > 0$.

(b) An astrophysicist is analyzing the energy density fluctuations in a nebula. These fluctuations are described by a ratio of terms involving a variable $x$. Simplify the following expression:

$\frac{(x^{-3})^2}{(x^4)^{-1}}$

Express your answer with a positive exponent.

(c) An engineer is designing a new composite material, and the stress distribution in a critical section is modeled by the following expression involving variables $x$ and $y$:

$\frac{(8x^6)^{\frac{2}{3}}}{16y^8}$

Simplify this expression.

A company is designing a new line of eco-friendly packaging for artisanal candles. The packaging is in the form of a right prism with a square base. The side length of the square base is $x$ cm, and the height of the prism is $h$ cm.

Given that the total external surface area of the box is $960$ cm$^2$, show that the height of the box, $h$, can be expressed as $h = \frac{240}{x} - \frac{x}{2}$.

Hence, show that the volume of the box, $V$, may be expressed as $V = 240x - \frac{1}{2}x^3$.

Sketch the graph of $V = 240x - \frac{1}{2}x^3$, for $0 \le x \le 20$.

Find an expression for $\frac{dV}{dx}$.

Find the value of $x$ which maximizes the volume of the box. Give your answer to three significant figures.

Hence, or otherwise, find the maximum possible volume of the box. Give your answer to three significant figures.

The company plans to fill these boxes with spherical candles. The design team assumes that they can calculate the exact number of candles in each box by dividing the calculated maximum volume of the box by the volume of a single candle and then rounding down to the nearest integer. Explain why the design team's assumption is incorrect.

A supermarket display manager is arranging cans in a triangular stack for a promotional event. Each layer of the stack is a row of cans, with the bottom layer having the most cans, and each subsequent layer having fewer cans, forming a triangular shape when viewed from the front. The top layer always has 1 can.

(a) Write down the number of cans in the bottom row of a display with

(i) 4 layers.

(ii) 5 layers.

(b) Mayumi notices that the number of cans in the bottom row of the displays forms an arithmetic sequence.

(i) Write down the common difference of this sequence.

(ii) Find an expression for the number of cans in the bottom row of a display with $n$ layers.

(c) The display manager wants to create a stack with 9 layers.

(i) Find the number of cans in the bottom row of this stack.

(ii) Calculate the total number of cans in a stack with 9 layers.

(d) Find an expression for the total number of cans in a stack with $n$ layers, giving your answer in its simplest form.

(e) The cans in the stack are either 'regular' or 'promotional'. The number of 'regular' cans in each layer is equal to the layer number (e.g., layer 1 has 1 regular can, layer 2 has 2 regular cans, and so on).

Write down the total number of 'regular' cans in a stack with 4 layers.

(f) Find and simplify an expression for the total number of 'regular' cans in a stack with $n$ layers.

(g) The total number of 'promotional' cans in a stack with $n$ layers is given by the expression $\frac{n(n-1)}{2}$.

Using both the given expression and your answer to part (f), find and simplify an expression for the total number of cans (regular and promotional) in a stack with $n$ layers.