Find the Domain h=-16t^2+64+80

Problem

h=−16*t^2+64*t+80

Solution

1. Identify the type of function. The expression h=−16*t^2+64*t+80 is a quadratic function, which is a type of polynomial.

2. Determine the mathematical domain. For any polynomial function, the variable t can be any real number because there are no square roots of negative numbers or divisions by zero.

3. Consider the physical context. In physics problems where h represents height and t represents time, the domain is usually restricted to the interval where time is non-negative (t≥0 and the object is at or above ground level (h≥0.

4. Find the roots of the equation to determine when the object hits the ground. Set h=0

0=−16*t^2+64*t+80

5. Factor the quadratic equation by dividing all terms by −16

0=t^2−4*t−5

6. Solve for t by factoring the trinomial.

0=(t−5)*(t+1)

7. Identify the valid time values. The roots are t=5 and t=−1 Since time cannot be negative in this context, the object is in the air from t=0 to t=5

Final Answer

Domain:0≤t≤5

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