Choose the option that gives (correct to two decimal places) the area of
the region which is bounded by the x-axis and by the graph of
y =(20(7 − x)(x − 3))/x^2 .
Options
A 0.47 B 1.31 C 2.09 D 3.76
E 5.12 F 6.55 G 8.20 H 9.46
To find the area of the region bounded by the x-axis and the given graph, we need to calculate the definite integral of the function with respect to x over the appropriate interval.
The graph of the function y = (20(7 − x)(x − 3))/x^2 intersects the x-axis at x = 0 and x = 7. We need to find the integral of the function between these two points.
First, let's simplify the function:
y = (20(7 − x)(x − 3))/x^2
= (20(7x - 3x - 21))/x^2
= (20(4x - 21))/x^2
= (80x - 420)/x^2
Now, we can find the integral of this function between x = 0 and x = 7:
∫[0,7] (80x - 420)/x^2 dx
To find the integral, we can split the expression into two separate integrals:
∫[0,7] (80x/x^2) dx - ∫[0,7] (420/x^2) dx
Simplifying the fractions:
∫[0,7] 80/x dx - ∫[0,7] 420/x^2 dx
Now, we can calculate each integral separately:
∫[0,7] 80/x dx = 80(ln|x|)[0,7]
= 80(ln|7|) - 80(ln|0|)
Note: ln|0| is undefined, so the lower limit of the integration must exclude x = 0.
∫[0,7] 420/x^2 dx = -420/x [0,7]
= -420/7 + 420/0
Note: 420/0 is undefined, so the upper limit of the integration must exclude x = 0.
Therefore, the area of the region bounded by the x-axis and the graph of y = (20(7 − x)(x − 3))/x^2 is equal to 80(ln|7|).
Now, we can calculate this value:
Area = 80(ln|7|)
Using a calculator, the value of ln|7| is approximately 1.95 (rounded to two decimal places).
Therefore, the area of the region is approximately:
Area ≈ 80(1.95) ≈ 156
Out of the given options, the closest value to 156 is 2.09 (Option C).
Therefore, the correct answer is Option C: 2.09.