To solve this problem, we need to establish a relationship between the speed [tex]\( v \)[/tex] and the maximum height [tex]\( h \)[/tex]. Based on the problem statement, we know that [tex]\( h \)[/tex] is proportional to the square of [tex]\( v \)[/tex], which means we can write this relationship as:
[tex]\[ h = k \cdot v^2 \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
### Step 1: Calculate the constant of proportionality [tex]\( k \)[/tex]
Using the given values [tex]\( v = 20 \)[/tex] and [tex]\( h = 8 \)[/tex]:
[tex]\[ 8 = k \cdot 20^2 \][/tex]
Now, solve for [tex]\( k \)[/tex]:
[tex]\[ 8 = k \cdot 400 \][/tex]
[tex]\[ k = \frac{8}{400} \][/tex]
[tex]\[ k = 0.02 \][/tex]
### Step 2: Find the maximum height [tex]\( h \)[/tex] when [tex]\( v = 35 \)[/tex]
We now have the constant [tex]\( k \)[/tex]. To find the maximum height [tex]\( h \)[/tex] when [tex]\( v = 35 \)[/tex], use the proportionality relationship:
[tex]\[ h = k \cdot 35^2 \][/tex]
Substitute [tex]\( k = 0.02 \)[/tex] and [tex]\( v = 35 \)[/tex]:
[tex]\[ h = 0.02 \cdot 35^2 \][/tex]
[tex]\[ h = 0.02 \cdot 1225 \][/tex]
[tex]\[ h = 24.5 \][/tex]
Therefore, the maximum height reached by the golf ball when [tex]\( v = 35 \)[/tex] is [tex]\( 24.5 \)[/tex] metres.
