Question

question
the area $a = a(x)$ of a right angled isosceles triangle is
$a = a(x)=\frac{1}{2}\times x\times x=\frac{1}{2}x^{2}$, where $x$ is the length of the two shortest sides.
which of the following statements are true? select all that apply.
select all that apply:
$x$ is a dependent variable
$a$ is dependent upon $x$.
$a$ is function of $x$.
$x$ is function of $a$.

Explanation:

Step1: Understand variable - type concept

In the equation $A(x)=\frac{1}{2}x^{2}$, the value of $A$ is determined by the value of $x$. So $x$ is an independent variable and $A$ is a dependent variable.

Step2: Analyze option A

$x$ is not a dependent variable. The value of $A$ changes as $x$ changes, not the other - way around. So option A is false.

Step3: Analyze option B

Since $A=\frac{1}{2}x^{2}$, the value of $A$ depends on the value of $x$. So option B is true.

Step4: Analyze option C

The notation $A = A(x)$ indicates that $A$ is a function of $x$, where for each input value of $x$ there is a corresponding output value of $A$ calculated by the formula $\frac{1}{2}x^{2}$. So option C is true.

Step5: Analyze option D

While we can solve the equation $A=\frac{1}{2}x^{2}$ for $x$ as $x=\sqrt{2A}$ (assuming $x\geq0$), the original relationship given is $A$ as a function of $x$. The problem statement is based on the initial form where $A$ is defined in terms of $x$, not the reverse - relationship. So option D is not in line with the original problem setup and is false.

Answer:

• B. $A$ is dependent upon $x$.
• C. $A$ is function of $x$.