Number Skills

[Maximum mark: 6]

Four cement bags labelled, "5 kg", were delivered to a customer. The customer measured each bag to check their weights and recorded the following:

1. 1. Find the mean of the customer's measurements.

   2. Calculate the percentage error between the mean and the stated,
      approximate weight of $5$ kg. [3]
      

2. Calculate $\sqrt{2.15^8-5.12^{-0.8}}$, giving your answer

   1. correct to the nearest integer;

   2. in the form $a\times10^k$, where $1 \leq a < 10$ and $k\in \mathbb{Z}$. [3]
      

[Maximum mark: 6]

The following diagram shows a rectangle with sides of length $7.6\times10^2$ cm and $1.5\times10^3$ cm.

1. Write down the area of the rectangle in the form $a\times10^k$, where
   $1 \leq a < 10$ and $k \in \mathbb{Z}$. [3]
   

Natalie estimates the area of the rectangle to be $1\hspace{0.1em}200\hspace{0.15em}000$ cm$^2$.

2. Find the percentage error in Natalie's estimate. [3]
   

[Maximum mark: 6]

Given $r = 2a - \dfrac{\sqrt{b}}{c}$, $a = 0.975$, $b = 4.41$ and $c = 35$,

1. calculate the value of $r$. [2]
   

Albert first writes $a$, $b$ and $c$ correct to one significant figure and then uses these values to estimate the value of $r$.

2. 1. Write down $a$, $b$ and $c$ each correct to one significant figure.

   2. Find Albert's estimate of the value of $r$. [2]
      

3. Calculate the percentage error in Albert's estimate of the value of $r$. [2]
   

[Maximum mark: 8]

A cuboid has the following dimensions: $\text{length} = 9.6\hspace{0.25em}$cm, $\text{width} = 7.4\hspace{0.25em}$cm, and $\text{height} = 5.2\hspace{0.25em}$cm, measured correct to the nearest millimetre.

1. Using these measurements, calculate the volume of the cuboid, in cm$^3$. Give your answer to two decimal places. [2]
   

The lower and upper bounds for the length of the cuboid can be expressed as $9.55 \leq l < 9.65$.

2. Write similar expressions for

   1. the width;

   2. the height. [2]
      

3. Hence, calculate the minimum volume of the cuboid. Give your answer to three significant figures. [2]
   

4. Write your answer to part (c) in the form $a\times10^k$, where $1 \leq a < 10$ and $k \in \mathbb{Z}$. [2]