Explore Measurement Concepts with Our Examples and Questions

Convert 315 degree to radian measure in terms of pi.

Solve each part step wise
a) A man walked for some distance at 8km/h, and fro an equaldistance at 5km/h . The total time he took was 3 1/4 hours. Findthe total distance walked.
b) A cyclist goes from one village to another at 28 km/h.He returns at 24 km/h. IF the return journey takes two hourslonger than the outward journey, what is the distance between thevillage?
c) If a piece wood is 5cm longer than a second piece and 3/4of the second piece is equal to 3/5 of the first , what is thelength of the second piece?
Note: Please Solve Step wise with variables.

The person who is 70 inches tall cast a shadow is 54 inches long. If a tree casts a shadow that is 28 feet long at te same time, then how tall is the tree to the nearest foot?

A package weighs 7.5 kg. There are 2.2 lb in 1 kg, and 16 oz in 1 lb. What is the weight of the package in ounces?

Henry eats 1/3 of a loaf of bread in 2 days. How many loaves of bread will he eat in 12 days?

$3\frac{5}{16}$ in decimal form

4 quarts (dry) to liters (to the nearest thousandth)

Use the formula $F=1.8(c)+32$ to find the Fahrenheit equivalent to 36.4 degrees celsius. F in the formula stands for Fahrenheit and C stands for Celsius

Draw, in standard position, the angle whose measure is given:
$\frac{4\mathrm{\pi <!--\; \pi \; -->}}{3}$

Draw, in standard position, the angle whose measure is given:
$-<!--\; -\; -->\frac{7\mathrm{\pi <!--\; \pi \; -->}}{2}$

Convert the unit of weight. Use 1 lb = 16 oz. Express your answer in exact simplest form.
$4\text{ lb}=\frac{\mathrm{◻<!--\; ◻\; -->}}{\mathrm{◻<!--\; ◻\; -->}}=\frac{\mathrm{◻<!--\; ◻\; -->}}{\mathrm{◻<!--\; ◻\; -->}}=\text{ oz}$

Assume I want to compute one of the angles of a right triangle doing n measurements of the sides with a ruler. In order to increase the precision I make several measurements. After that I compute $\tan <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}$ for each of the measurements:
${\tan }^{(i)}<!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}=\frac{{\text{opposite side}}^{(i)}}{{\text{adjacent side}}^{(i)}}$
where (i) denotes the number of the measurement.
Which formulae out of two for computing the average would be more precise?
${\mathrm{\theta <!--\; \theta \; -->}}_1={\tan }^{-<!--\; -\; -->1}<!--\; \; -->(\frac{1}{n}\underset{i}{\sum <!--\; \sum \; -->}{\tan }^{(i)}<!--\; \; -->\mathrm{\theta <!--\; \theta \; -->})$
${\mathrm{\theta <!--\; \theta \; -->}}_2=\frac{1}{n}\underset{i}{\sum <!--\; \sum \; -->}{\tan }^{-<!--\; -\; -->1}<!--\; \; -->({\tan }^{(i)}<!--\; \; -->\mathrm{\theta <!--\; \theta \; -->})$
Assume I have a computer which will be doing the tan−1 calculation. I am really interested in the proof as well.

If $f:<!--\; :\; -->I\subseteq <!--\; \subseteq \; -->\mathbb{R}\to <!--\; \to \; -->\mathbb{R}$ is a Lebesgue-measurable function ($I$ is a closed interval), $\mathrm{\mu <!--\; \mu \; -->}$ is the Lebesgue measure, my question is whether the limit
$\underset{u\to <!--\; \to \; -->u_0^{-<!--\; -\; -->}}{lim}\mathrm{\mu <!--\; \mu \; -->}(f^{-<!--\; -\; -->1}((u,u_0]))=0.$
I don't really know how to justify the value of this limit, if it is 0 or possibly $\mathrm{\mu <!--\; \mu \; -->}(f^{-<!--\; -\; -->1}(\{u_0\}))$. I would appreciate your help.

1. Convert 37 degrees celsius to Fahrenheit. Round to the nearest tenth of a degree
2. Seven (7) ml of a pain reliever contains 160 mg of the active ingredient. Ann needs 400 mg of the active ingredient. How many ml of the pain reliever should Ann take? State the unit

Properly showing that uncountable sum of measures finite nonzero sets is infinite, given they are subsets of $X$ of finite measure.
In a question where $(X,\sum <!--\; \sum \; -->,\mathrm{\mu <!--\; \mu \; -->})$ is a finite measure space, I am asked to show that the set $\{x|\mathrm{\mu <!--\; \mu \; -->}(\{x\}>0)\}$ is countable at most.
While intuitively this has to be true, because as long as it is countable any series of nonzero measures of sets must converge and this can happen for geometric sequences of measures, but for an uncountable sum, this "can't" be the case, but how to show that- I can't seem to understand.
Firstly, is this the way to solve it? Trying to reach a contradiction, or maybe uncountable sum of elements is generally meaningless as an expression? I'm not sure, what

I am trying to get my head around angles in projective geometry.
I understand (more or less) the cross ratio and that it can be seen as an distance measurement. (for example in the Beltrami Cayley Klein model of hyperbolic geometry)
But then there is its dual where the cross ratio is some kind of angle measurement of lines?
I just draw a blank here
which lines are mend here? can anybody give some light?

How can I show using the monotone convergence theorem that the Lebesgue integral
${\int <!--\; \int \; -->}_0^{\mathrm{\infty <!--\; \infty \; -->}}\frac{\sin <!--\; \; -->(x)}{x}dx$
does not exist?

Draw, in standard position, the angle whose measure is given:
$-<!--\; -\; -->{315}^{\circ <!--\; \circ \; -->}$

I read in an old book the following example of a measure: For the set $M=\mathbb{Q}\cap <!--\; \cap \; -->[0,1]$ denote with $S$ the set system of subsets of $M$ of the form $\mathbb{Q}\cap <!--\; \cap \; -->I$, where $I$ is any interval in [0,1]. Let us define the function $\mathrm{\mu <!--\; \mu \; -->}:S\to <!--\; \to \; -->\mathbb{R}$ as follows: for any set $A\in <!--\; \in \; -->S$ of the form $A=\mathbb{Q}\cap <!--\; \cap \; -->I$ we set $\mathrm{\mu <!--\; \mu \; -->}(A)=\mathrm{\ell <!--\; \ell \; -->}(I)=b-<!--\; -\; -->a.$.
Then it said without proof that μ is finitely additive, but not σ-additive.
As I did not get why I tried to prove it by myself and I tried to show that $S$ is a semi-ring, I guess that is important before I start with the other proof.
We have $\mathrm{\varnothing <!--\; \varnothing \; -->}\in <!--\; \in \; -->\mathbb{Q}$ and furthermore $(\mathbb{Q}\cap <!--\; \cap \; -->I_1)\cap <!--\; \cap \; -->(\mathbb{Q}\cap <!--\; \cap \; -->I_2)=\mathbb{Q}\cap <!--\; \cap \; -->I_1\cap <!--\; \cap \; -->I_2$ and the union of two closed intervals is either an interval or the disjoint union of two intervals. Then $(\mathbb{Q}\cap <!--\; \cap \; -->I_1)\setminus <!--\; \setminus \; -->(\mathbb{Q}\cap <!--\; \cap \; -->I_2)$ is also an interval or the disjoint union of two intervals.
Now the proof. I do not quite understand how it cannot be $\mathrm{\sigma <!--\; \sigma \; -->}$-additive. Does it have something in common with Cantor sets? I don´t know how to start the proof here. Any help or explanation (maybe an idea for the beginning of a proof) is appreciated. If there is a proof...

I was wondering: a measure $\mathrm{\mu <!--\; \mu \; -->}$ is a function that takes a set of numbers $S\in <!--\; \in \; -->{\mathbb{R}}^n$ and assign a non-negative number to it. I'm summary: $\mathrm{\mu <!--\; \mu \; -->}:S\to <!--\; \to \; -->{\mathbb{R}}_{+}$Does any of you know if there are measures like this:
$\mathrm{\mu <!--\; \mu \; -->}:S\to <!--\; \to \; -->{\mathbb{R}}_{+}^2$
The final aim is being able to distinguish between countable dense sets and non dense sets, so a subset $S\in <!--\; \in \; -->\mathbb{Q}$ can have a measure greater than zero if it's dense, at least in one of the dimensions of $\mathrm{\mu <!--\; \mu \; -->}$.
Many thanks in advance!