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What is the average value of sqrtx on the interval [5,12] ? Choose 1 answer: (A) (2)/(21)*(sqrt(12^(3))-sqrt(5^(3))) (B) (2)/(21)*(root(3)(12^(2))-root(3)(5^(2))) (C) (1)/(2)*(sqrt12-sqrt5) (D) (1)/(2)*(sqrt12+sqrt5)

What is the average value of x \sqrt{x} on the interval [5,12] [5,12] ?\newlineChoose 11 answer:\newline(A) 221⋅(123−53) \frac{2}{21} \cdot\left(\sqrt{12^{3}}-\sqrt{5^{3}}\right) \newline(B) 221⋅(1223−523) \frac{2}{21} \cdot\left(\sqrt[3]{12^{2}}-\sqrt[3]{5^{2}}\right) \newline(C) 12⋅(12−5) \frac{1}{2} \cdot(\sqrt{12}-\sqrt{5}) \newline(D) 12⋅(12+5) \frac{1}{2} \cdot(\sqrt{12}+\sqrt{5})

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Full solution

Q. What is the average value of x \sqrt{x} on the interval [5,12] [5,12] ?\newlineChoose 11 answer:\newline(A) 221⋅(123−53) \frac{2}{21} \cdot\left(\sqrt{12^{3}}-\sqrt{5^{3}}\right) \newline(B) 221⋅(1223−523) \frac{2}{21} \cdot\left(\sqrt[3]{12^{2}}-\sqrt[3]{5^{2}}\right) \newline(C) 12⋅(12−5) \frac{1}{2} \cdot(\sqrt{12}-\sqrt{5}) \newline(D) 12⋅(12+5) \frac{1}{2} \cdot(\sqrt{12}+\sqrt{5})
  1. Set Up Integral: To find the average value of a function f(x)f(x) on the interval [a,b][a, b], we use the formula for the average value of a function on an interval, which is given by:\newlineAverage value = 1(b−a)∫abf(x) dx\frac{1}{(b-a)} \int_{a}^{b} f(x) \, dx\newlineHere, f(x)=xf(x) = \sqrt{x}, a=5a = 5, and b=12b = 12.
  2. Calculate Integral: First, we need to set up the integral to find the average value of x\sqrt{x} from x=5x = 5 to x=12x = 12.\newlineAverage value = 1(12−5)×∫512x dx\frac{1}{(12-5)} \times \int_{5}^{12} \sqrt{x} \, dx
  3. Evaluate Antiderivative: Now, we calculate the integral of x\sqrt{x} from 55 to 1212. The antiderivative of x\sqrt{x} is 23x32\frac{2}{3}x^{\frac{3}{2}}, so we will evaluate this from 55 to 1212. ∫512x dx=[23x32]512\int_{5}^{12} \sqrt{x} \, dx = \left[\frac{2}{3}x^{\frac{3}{2}}\right]_{5}^{12}
  4. Simplify Expression: Next, we plug in the upper and lower limits of the integral into the antiderivative.\newline=23(1232)−23(532)= \frac{2}{3}(12^{\frac{3}{2}}) - \frac{2}{3}(5^{\frac{3}{2}})
  5. Divide by Interval Length: Now, we simplify the expression by calculating the powers and multiplying by the coefficients.\newline=23(123)−23(53)= \frac{2}{3}(\sqrt{12^3}) - \frac{2}{3}(\sqrt{5^3})\newline=23(1728)−23(125)= \frac{2}{3}(\sqrt{1728}) - \frac{2}{3}(\sqrt{125})\newline=23(1212)−23(55)= \frac{2}{3}(12\sqrt{12}) - \frac{2}{3}(5\sqrt{5})
  6. Distribute Coefficients: We then divide the result by (12−5)(12-5), which is the length of the interval.\newlineAverage value = 17\frac{1}{7} * (23)(1212)−(23)(55)\left(\frac{2}{3}\right)(12\sqrt{12}) - \left(\frac{2}{3}\right)(5\sqrt{5})
  7. Compare Answer Choices: Simplify the expression by distributing the 17\frac{1}{7} into the bracket.\newlineAverage value = 221(1212)−221(55)\frac{2}{21}(12\sqrt{12}) - \frac{2}{21}(5\sqrt{5})
  8. Correct Simplification: Now, we look at the answer choices to see which one matches our expression.\newlineThe correct answer choice should be equivalent to (221)(1212)−(221)(55)(\frac{2}{21})(12\sqrt{12}) - (\frac{2}{21})(5\sqrt{5}).
  9. Correct Simplification: Now, we look at the answer choices to see which one matches our expression.\newlineThe correct answer choice should be equivalent to (221)(1212)−(221)(55)(\frac{2}{21})(12\sqrt{12}) - (\frac{2}{21})(5\sqrt{5}).By comparing the answer choices, we can see that none of them match our expression exactly. However, we can simplify our expression further to see if it matches any of the given options.\newline(221)(1212)−(221)(55)(\frac{2}{21})(12\sqrt{12}) - (\frac{2}{21})(5\sqrt{5}) can be rewritten as:\newline(221)(123)−(221)(53)(\frac{2}{21})(\sqrt{12^3}) - (\frac{2}{21})(\sqrt{5^3})
  10. Correct Simplification: Now, we look at the answer choices to see which one matches our expression.\newlineThe correct answer choice should be equivalent to (221)(1212)−(221)(55)(\frac{2}{21})(12\sqrt{12}) - (\frac{2}{21})(5\sqrt{5}).By comparing the answer choices, we can see that none of them match our expression exactly. However, we can simplify our expression further to see if it matches any of the given options.\newline(221)(1212)−(221)(55)(\frac{2}{21})(12\sqrt{12}) - (\frac{2}{21})(5\sqrt{5}) can be rewritten as:\newline(221)(123)−(221)(53)(\frac{2}{21})(\sqrt{12^3}) - (\frac{2}{21})(\sqrt{5^3})We realize that there has been a mistake in the simplification process. The correct simplification should be:\newline(221)(123)−(221)(53)(\frac{2}{21})(\sqrt{12^3}) - (\frac{2}{21})(\sqrt{5^3})\newline= (221)(12⋅12⋅12)−(221)(5⋅5⋅5)(\frac{2}{21})(\sqrt{12}\cdot\sqrt{12}\cdot\sqrt{12}) - (\frac{2}{21})(\sqrt{5}\cdot\sqrt{5}\cdot\sqrt{5})\newline= (221)(1212)−(221)(55)(\frac{2}{21})(12\sqrt{12}) - (\frac{2}{21})(5\sqrt{5})\newlineThis is the same as our previous step, and it seems we have not made any progress in simplifying it to match the answer choices. We need to re-evaluate our steps to find the correct simplification.

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