To find the value of [tex]\( b \)[/tex] in the equation of an ellipse given the distance to a vertex and a focus, we start with the following pieces of information:
- The distance to the vertex along the major axis is represented by [tex]\( a \)[/tex]. Given [tex]\( a = 20 \)[/tex].
- The distance from the center to the focus is represented by [tex]\( c \)[/tex]. Given [tex]\( c = 16 \)[/tex].
The relationship between [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] for an ellipse is given by:
[tex]\[ c^2 = a^2 - b^2 \][/tex]
First, we calculate [tex]\( a^2 \)[/tex]:
[tex]\[ a^2 = 20^2 = 400 \][/tex]
Next, we calculate [tex]\( c^2 \)[/tex]:
[tex]\[ c^2 = 16^2 = 256 \][/tex]
Using the relationship [tex]\( c^2 = a^2 - b^2 \)[/tex], we solve for [tex]\( b^2 \)[/tex]:
[tex]\[ b^2 = a^2 - c^2 \][/tex]
[tex]\[ b^2 = 400 - 256 \][/tex]
[tex]\[ b^2 = 144 \][/tex]
To find [tex]\( b \)[/tex], we take the square root of [tex]\( b^2 \)[/tex]:
[tex]\[ b = \sqrt{144} = 12 \][/tex]
So, the value of [tex]\( b \)[/tex] in the equation is:
[tex]\[ 12 \][/tex]
Hence, the correct answer is [tex]\( \boxed{12} \)[/tex].
