Respuesta :
we have the equation
[tex]576x^2+625y^2=360,000[/tex]Simplify
Divide both sides by 360,000
so
[tex]\begin{gathered} \frac{576}{360,000}x^2+\frac{625}{360,000}y^2=\frac{360,000}{360,000} \\ \\ \\ \frac{x^2}{625}+\frac{y^2}{576}=1 \end{gathered}[/tex]Remember that
625=25^2
576=24^2
substitute
[tex]\frac{x^2}{25^2}+\frac{y^2}{24^2}=1[/tex]so
a^2=25 and b^2=24
we have the function f(x)
[tex]f(x)=\frac{b}{a}\sqrt[]{a^2-x^2}[/tex]the point A is (15,f(15))
Calculate f(15)
[tex]f(15)=\frac{\sqrt[]{24}}{\sqrt[]{25}}\sqrt[]{25^2-15^2}[/tex][tex]f(15)=\frac{2\sqrt[]{6}}{5}\sqrt[]{400}[/tex]simplify
[tex]f(15)=8\sqrt[]{6}[/tex]the point A is
[tex]A(15,\text{ 8}\sqrt[]{6})[/tex]Find the point B
B(20,f(20)
Calculate f(20)
[tex]f(20)=\frac{\sqrt[]{24}}{\sqrt[]{25}}\sqrt[]{25^2-20^2}[/tex][tex]f(20)=6\sqrt[]{6}[/tex]the point B is
[tex]B(20,6\sqrt[]{6})[/tex]Determine the distance between A and B
Applying the formula to calculate the distance between two points
[tex]d=\sqrt[\square]{(y2-y1)^2+(x2-x1)^2}[/tex]substitute the given values of A and B
[tex]d_{AB}=\sqrt[\square]{(6\sqrt[]{6}-8\sqrt[]{6})^2+(20-15)^2}[/tex][tex]\begin{gathered} d_{AB}=\sqrt[\square]{24+25} \\ d_{AB}=7 \end{gathered}[/tex]therefore
