Step-by-step explanation:
Given
[tex]h(t)=(139kt)^2-69t+80[/tex]
For roots [tex]h(t)=0[/tex]
[tex](139kt)^2-69t+80=0\\t=\dfrac{69\pm\sqrt{(-69)^2-4\times (139k)^2(80)}}{2\times 139}\\[/tex]
[tex]t=\dfrac{69\pm\sqrt{4761-6182720k^2}}{278}[/tex]
Larger root: [tex]t=\dfrac{69+\sqrt{4761-6182720k^2}}{278}[/tex]
smaller root: [tex]t=\dfrac{69-\sqrt{4761-6182720k^2}}{278}[/tex]
For exactly one root D=0
i.e. [tex]4761-6182720k^2=0\\\\k=\dfrac{69}{139\times 4\times 2\times \sqrt{5}}\\k=0.0277[/tex]
