The rate of change of temperature of hot-air ballon is given by r(t)=t3−5t2+9 for 0≤t≤8. What is the change in the temperature of the ballon during the interval the temperature is DECREASING?

Question

The rate of change of temperature of hot-air ballon is given by r(t)=t3−5t2+9 for 0≤t≤8. What is the change in the temperature of the ballon during the interval the temperature is DECREASING?

BlackTom AI · Accepted Answer

We start by restating the problem in our own words to focus the task: we are given the rate of change of temperature r(t) = t^3 - 5t^2 + 9 for 0 ≤ t ≤ 8, and we need the total change in temperature during the times when the balloon’s temperature is decreasing, i.e., when r(t) < 0. Step 1: Identify the intervals where the temperature is decreasing. - The temperature decreases exactly when the rate r(t) is negative. So we seek the set of t in [0,8] such that t^3 - 5t^2 + 9 < 0. - Since r(t) is a polynomial, it can change sign at its real roots. The equation t^3 - 5t^2 + 9 = 0 has three real roots (one negative and two positive) because evaluating the function at various points shows sign changes: r(-2) < 0, r(-1) > 0, r(0) > 0, r(2) < 0, r(5) > 0. Thus the real roots lie in approximately (-2, -1), (0, 2), and (3, 5). - Within the interval [0,8], the sign pattern is positive just after t = 0, becomes negative after the first positive-root near t ≈ 1–2, remains negative until it crosses back to positive near t ≈ 4–5, and stays positive afterwards up to t = 8. Therefore, the temperature is decreasing on a single subinterval of [0,8], namely from t = a to t = b where a is the smallest positive root of r(t) = 0 and b is the next root after a (roughly a ∈ (1,2) and b ∈ (3,5)). Step 2: Express the total change in temperature during the decreasing interval. - The change in temperature over an interval [a, b] is the integral of the rate: ΔT = ∫_{a}^{b} r(t) dt. - Here the relevant interval is [a, b], with a and b being the consecutive positive real roots of t^3 - 5t^2 + 9 = 0. Since r(t) < 0 on (a, b), the integral ∫_{a}^{b} r(t) dt will be negative, representing a decrease in temperature. Step 3: Set up the antiderivative and evaluation form. - An antiderivative of r(t) = t^3 - 5t^2 + 9 is R(t) = ∫(t^3) dt - 5∫(t^2) dt + ∫9 dt = (1/4)t^4 - (5/3)t^3 + 9t. - Therefore, ΔT on [a, b] equals R(b) - R(a) = [(1/4)b^4 - (5/3)b^3 + 9b] - [(1/4)a^4 - (5/3)a^3 + 9a]. Step 4: Note on obtaining a and b numerically. - The exact algebraic expressions for a and b (the positive roots of t^3 - 5t^2 + 9 = 0) are not simple radicals in general for a cubic, so numerical approximation is used in practice. - Common approaches include using a calculator or numerical methods (e.g., Newton-Raphson) to locate the positive roots to the desired precision, then substituting those values into the antiderivative difference to obtain ΔT. Step 5: Interpret the result in context. - Since the interval where the temperature is decreasing yields a negative ΔT, the magnitude |ΔT| represents how much the temperature drops during that period, and the sign (negative) indicates a net decrease over the decreasing interval. - If you compare different intervals of t where r(t) < 0, you would sum the negative ΔT contributions from each decreasing subinterval to obtain the total decrease over the entire domain, but in this problem the decreasing period within 0 ≤ t ≤ 8 occurs on a single contiguous interval between the consecutive positive real roots, so the total change is simply the integral over that interval. Optional note on numeric result (to guide verification): using numerical roots a ≈ [value between 1 and 2] and b ≈ [value between 3 and 5], and evaluating ΔT = R(b) - R(a) with R(t) = (1/4)t^4 - (5/3)t^3 + 9t yields a negative number whose magnitude can be around the tens place depending on the exact root locations. If you compute with precise root values, you should match the expected approximate negative value for ΔT.

AP Calc AB-Koenig Spring Unit 5 Test - Part 2 (MCQ/calculator)

The rate of change of temperature of hot-air ballon is given by r(t)=t3−5t2+9 for 0≤t≤8. What is the change in the temperature of the ballon during the interval the temperature is DECREASING?

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