A penguin leaps out of the water while swimming. The height y (in feet) of a porpoising penguin can be modeled by y = -16x^2 + 4.8x, where x is the time (in seconds) since the peng... A penguin leaps out of the water while swimming. The height y (in feet) of a porpoising penguin can be modeled by y = -16x^2 + 4.8x, where x is the time (in seconds) since the penguin leaped out of the water. Find the roots of the equation when y = 0.
Understand the Problem
The problem describes the height of a porpoising penguin as a function of time using a quadratic equation. It asks to find the roots of the equation when y = 0, which represents finding the times when the penguin is at the water level (height = 0). This involves solving the quadratic equation -16x^2 + 4.8x = 0 for x.
Answer
The roots are $x = 0$ and $x = 0.3$.
Answer for screen readers
The roots of the equation are $x = 0$ and $x = 0.3$.
Steps to Solve
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Set up the equation We are given the equation $y = -16x^2 + 4.8x$. We need to find the roots when $y = 0$. So we have $-16x^2 + 4.8x = 0$.
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Factor out the common factor Both terms have $x$ in them, and also have a common numerical factor. Factoring out $-1.6x$ from both terms gives $-1.6x(10x - 3) = 0$. We can factor out simply $x$ as well to get: $x(-16x + 4.8) = 0$.
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Solve for x Using the zero product property, either $x = 0$ or $-16x + 4.8 = 0$.
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Solve the second equation If $-16x + 4.8 = 0$, then $16x = 4.8$, so $x = \frac{4.8}{16}$.
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Simplify the fraction Simplify the fraction $x = \frac{4.8}{16}$ by multiplying the numerator and denominator by 10 to remove the decimal: $x = \frac{48}{160}$. Both 48 and 160 are divisible by 16, so $x = \frac{3}{10} = 0.3$.
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State the roots The roots are $x = 0$ and $x = 0.3$.
The roots of the equation are $x = 0$ and $x = 0.3$.
More Information
The roots represent the times when the penguin is at water level. The first root, $x = 0$, represents the time when the penguin initially leaps out of the water. The second root, $x = 0.3$, represents the time when the penguin comes back down to the water.
Tips
A common mistake is to forget to factor out the $x$ and only solve $-16x + 4.8 = 0$, missing the root $x = 0$. Another common mistake is to make an error in simplifying the fraction.
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