Respuesta :
The given ellipse is
[tex]576x^2+625y^2=360,000[/tex]Where A(15, f(15)), B(20, f(20)).
First, we find f(15) and f(20) by evaluating the given expression
[tex]\begin{gathered} f(15)=576(15)^2+625y^2=360,000 \\ 576\cdot225+625y^2=360,000 \\ 129,600+625y^2=360,000 \\ 625y^2=360,000-129,600 \\ 625y^2=230,400 \\ y^2=\frac{230,400}{625} \\ y^2=368.64 \\ y=\sqrt[]{368.64} \\ y=19.2 \end{gathered}[/tex]We use the same process to find f(20).
[tex]\begin{gathered} f(20)=576(20)^2+625y^2=360,000 \\ 576\cdot400+625y^2=360,000 \\ 625y^2=360,000-230,400 \\ x^2=\frac{129,600}{576} \\ x=\sqrt[]{225}=15 \\ \end{gathered}[/tex]So, the points are A(15, 19.2) and B(20, 15). To find the distance between these points, we have to use the distance formula
[tex]\begin{gathered} d_{AB}=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ d_{AB}=\sqrt[]{(20-15)^2+(15-19.2)^2} \\ d_{AB}=\sqrt[]{5^2+(-4.2)^2}=\sqrt[]{25+17.64}=\sqrt[]{42.64} \\ d_{AB}\approx6.5 \end{gathered}[/tex]