SOLUTION
Given the question in the question, the following are the solution steps to answer the question.
STEP 1: Write the relationship between the banking angle and the radius
[tex]\begin{gathered} \text{Let the banking angle be represented with a} \\ \text{Let the radius be represented with r} \\ \\ \text{From the statement, the banking angle varies inversely as the cycle's turning radius, this implies that:} \\ a\propto\frac{1}{r} \end{gathered}[/tex]
STEP 2: Solve for the constant of variation from the inverse variation in step 1
[tex]\begin{gathered} a\propto\frac{1}{r} \\ \text{Introducing the constant k} \\ a=\frac{k}{r} \\ By\text{ cross multiplication,} \\ ar=k \end{gathered}[/tex]
STEP 3: Get the value of k for the given radius and banking angle
[tex]\begin{gathered} k=ar \\ a=30,r=4 \\ k=30\times4=120 \end{gathered}[/tex]
STEP 4: Get the banking for the given turning radius
[tex]\begin{gathered} k=ar \\ \text{k is constant and is 120} \\ r=7.5,a=\text{?} \\ By\text{ substitution,} \\ 120=a\times7.5 \\ \text{Divide both sides by 7.5} \\ \frac{120}{7.5}=\frac{a\times7.5}{7.5} \\ 16=a \\ a=16^{\circ} \end{gathered}[/tex]
Hence, the banking angle for the given turning radius is 16 degrees
