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Answer :
Final answer:
The cost of charging the automotive battery for 4 hours at a constant current of 3 ampere with a terminal voltage of 10 + t/2 V is $42.
Explanation:
To calculate the cost of charging the automotive battery, we first need to find the total energy consumed during the charging process. The energy (E) consumed by the battery can be calculated using the formula: E = P * t, where P is the power (in watts) and t is the time (in hours). Since power (P) is given by the product of current (I) and voltage (V), we have P = I * V. Substituting the given values, we get P = 3 * (10 + t/2) = 30 + 1.5t watts.
Next, we find the total energy consumed by integrating the power function over time. Integrating from 0 to 4 hours, we get E = ∫(30 + 1.5t) dt from 0 to 4, which equals [(30t + 0.75t^2)/2] evaluated from 0 to 4, resulting in 42 watt-hours.
Finally, to find the cost, we multiply the total energy consumed (in watt-hours) by the cost per watt-hour. If the cost per watt-hour is $1, then the cost of charging the battery for 4 hours would be $42.
The given statement "A constant current of 3 ampere for 4 hours is required to charge an automotive battery. If the terminal voltage is 10+t/2 V, where t is in hours, how much does the charging cost?" is true because the total cost is calculated by integrating the power function over time and multiplying the total energy consumed by the cost per watt-hour, resulting in a final cost of $42 (Option A).
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