Let's work through the given problems step by step:
### Part (a)
You are asked to find the value of [tex]\( s \)[/tex] when [tex]\( a = 0.9 \)[/tex] and [tex]\( t = 4 \)[/tex].
Given the formula:
[tex]\[ s = \frac{1}{2} a t^2 \][/tex]
We substitute the given values [tex]\( a = 0.9 \)[/tex] and [tex]\( t = 4 \)[/tex] into the formula:
[tex]\[ s = \frac{1}{2} \times 0.9 \times 4^2 \][/tex]
First, compute [tex]\( 4^2 \)[/tex] (the square of 4):
[tex]\[ 4^2 = 16 \][/tex]
Next, multiply [tex]\( 0.9 \)[/tex] by 16:
[tex]\[ 0.9 \times 16 = 14.4 \][/tex]
Now, multiply this result by [tex]\( \frac{1}{2} \)[/tex] (which is the same as dividing by 2):
[tex]\[ s = \frac{1}{2} \times 14.4 = 7.2 \][/tex]
So, the value of [tex]\( s \)[/tex] is:
[tex]\[ s = 7.2 \][/tex]
### Part (b)
We need to rearrange the formula to solve for [tex]\( t \)[/tex] in terms of [tex]\( s \)[/tex] and [tex]\( a \)[/tex].
Starting with the original formula:
[tex]\[ s = \frac{1}{2} a t^2 \][/tex]
First, multiply both sides by 2 to eliminate the fraction:
[tex]\[ 2s = a t^2 \][/tex]
Next, divide both sides by [tex]\( a \)[/tex] to solve for [tex]\( t^2 \)[/tex]:
[tex]\[ t^2 = \frac{2s}{a} \][/tex]
Finally, take the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \sqrt{\frac{2s}{a}} \][/tex]
This gives the formula for [tex]\( t \)[/tex] in terms of [tex]\( s \)[/tex] and [tex]\( a \)[/tex]:
[tex]\[ t = \sqrt{\frac{2s}{a}} \][/tex]
Using the values from part (a) where [tex]\( s = 7.2 \)[/tex] and [tex]\( a = 0.9 \)[/tex], you can verify that:
[tex]\[ t = \sqrt{\frac{2 \times 7.2}{0.9}} \][/tex]
So, rearranging and evaluating confirms that [tex]\( t = 4.0 \)[/tex]. Hence,
the value of [tex]\( t \)[/tex] in terms of [tex]\( s \)[/tex] and [tex]\( a \)[/tex] is:
[tex]\[ t=\sqrt{\frac{2s}{a}}=4.0. \][/tex]
Therefore [tex]\( t = 4.0 \)[/tex].
