Question

The volume \[\left( v \right)\] of a sphere varies directly as the cube of its diameter \[\left( d \right)\] . How do you write this statement in algebraic language, using an equation with the variables \[c,v,{\text{and }}d\] ?

Answer 1

Hint:Algebraic expressions are expressions which are made up of variables and constants along with mathematical algebraic operations which are addition, subtraction, multiplication, division etc. Also when $a$ is proportional to $b$, it’s written mathematically as:
$a \propto b$ So using the above information we can solve the given question.

Complete step by step solution:
Given statements:
\[{\text{The volume}}\left( v \right){\text{of a sphere varies directly as the cube of its
diameter}}\left( d \right)........................\left( i \right)\]
Now we know that in mathematics the algebraic expressions are those expressions which are made up of variables and constants along with mathematical algebraic operations which are addition, subtraction, multiplication, division etc.
Algebraic expressions are widely useful and popular since it represents the value of an expression for all of the values a variable can take on.
Now according to the basic definition of proportionality when two terms are proportional we can write that as:
$a \propto b...................................\left( {ii} \right)$
Also the statement cube of its diameter represents the term:
${d^3}....................................\left( {iii} \right)$
Now using (ii) and (iii) we can write (i) as:
$v \propto {d^3}................................\left( {iv} \right)$
Now in this particular question we have to find an equation such that the proportionality sign has to be removed.
In order to remove a proportionality sign we should multiply it with a constant $c$ in the RHS.
Such that we can write:
$
v \propto {d^3} \\
v = c{d^3}................................\left( v \right) \\
$
Therefore the corresponding algebraic expression using an equation with the variables \[c,v,{\text{and }}d\] would be: $v = c{d^3}$

Note: While writing an algebraic expression some of the commonly used words and their corresponding mathematical operations are given below:
1. Addition : plus, sum, more than
2. Subtraction : subtracted, minus, less than, decreased by
3. Multiplication : times, product
4. Division : divided, quotient
Also while making an algebraic expression one must take care that the words and the operations used are mathematically correct or not.