A robotics engineer is programming a robot to navigate a series of platforms. The robot's vertical position, P(t), in meters, at time 't' seconds, is described by the function P(t) = (t - 2)(t - 8). If the robot needs to be at ground level (P(t) = 0) to transition between platforms, at what times does this occur?
- AAt t = 8 seconds only
- BAt t = 2 seconds only
- CAt t = 2 seconds and t = 8 seconds
- DAt t = 0 seconds and t = 10 seconds
Show answer & explanationAnswer & explanation
Correct answer: C. At t = 2 seconds and t = 8 seconds
To find the times when the robot is at ground level, we set P(t) = 0. The function is given in factored form: P(t) = (t - 2)(t - 8). According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero: t - 2 = 0 or t - 8 = 0. Solving these linear equations gives t = 2 or t = 8. Thus, the robot is at ground level at 2 seconds and 8 seconds.
Why the other options are wrong
- A. This only identifies one of the two correct times.
- B. This only identifies one of the two correct times.
- D. These values are incorrect and do not satisfy the equation P(t)=0.
Zero Product Property
The Zero Product Property states that if the product of two or more factors is zero, then at least one of the individual factors must be zero. It is commonly used to solve quadratic equations in factored form.
- If A * B = 0, then A = 0 or B = 0 (or both).
- This property is fundamental for solving equations by factoring.
- It helps find the x-intercepts (roots/zeros) of polynomial functions.
Memory trick: Factors to zero, means each part's a hero!