the equation - 16x² + 10x + 15 = y represents the height y, in feet, of a flotation device above the water…

the equation - 16x² + 10x + 15 = y represents the height y, in feet, of a flotation device above the water after x seconds. the linear term represents the initial velocity. the constant term represents the initial height.\na. if the initial velocity is 0, when should the flotation device land in the water?\nb. if the initial height is 0, when should the flotation device be in the water?\na. the flotation device should land in the water at the time □ s.\n(type an integer or decimal rounded to two decimal places as needed.)

the equation - 16x² + 10x + 15 = y represents the height y, in feet, of a flotation device above the water after x seconds. the linear term represents the initial velocity. the constant term represents the initial height.\na. if the initial velocity is 0, when should the flotation device land in the water?\nb. if the initial height is 0, when should the flotation device be in the water?\na. the flotation device should land in the water at the time □ s.\n(type an integer or decimal rounded to two decimal places as needed.)

Answer

Explanation:

Step1: Set up the equation for part a

When initial velocity is 0, the linear - term is 0. So the equation is $y=-16x^{2}+15$. When the device lands in the water, $y = 0$. So we have $-16x^{2}+15 = 0$.

Step2: Solve the quadratic equation for part a

Rearrange the equation to $16x^{2}=15$, then $x^{2}=\frac{15}{16}$. Taking the square - root of both sides, $x=\pm\sqrt{\frac{15}{16}}=\pm\frac{\sqrt{15}}{4}$. Since time $x\geq0$, $x=\frac{\sqrt{15}}{4}\approx0.97$.

Step3: Set up the equation for part b

When initial height is 0, the constant term is 0. So the equation is $y=-16x^{2}+10x$. When the device lands in the water, $y = 0$. So we have $-16x^{2}+10x = 0$.

Step4: Solve the quadratic equation for part b

Factor out an $x$: $x(-16x + 10)=0$. This gives two solutions: $x = 0$ (corresponds to the starting time) and $-16x+10 = 0$. Solving $-16x+10 = 0$ for $x$, we get $16x = 10$, so $x=\frac{10}{16}=\frac{5}{8}=0.625$.

Answer:

a. $0.97$ b. $0.625$