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Certainly! Let's solve the given equation step-by-step:
Step 1: Understand the given equation
The equation provided is:
[tex]\[ \frac{\int (x - 10) \, dx}{90^\circ} = 20^\circ \][/tex]
Step 2: Convert degrees to radians
We need to work with radians instead of degrees for the integration. The conversion factor between degrees and radians is:
[tex]\[ 1^\circ = \frac{\pi}{180} \text{ radians} \][/tex]
So, we convert [tex]\( 20^\circ \)[/tex] to radians:
[tex]\[ 20^\circ = 20 \times \frac{\pi}{180} = \frac{20\pi}{180} = \frac{\pi}{9} \text{ radians} \][/tex]
Similarly, [tex]\( 90^\circ \)[/tex] in radians is:
[tex]\[ 90^\circ = 90 \times \frac{\pi}{180} = \frac{90\pi}{180} = \frac{\pi}{2} \text{ radians} \][/tex]
Step 3: Re-write the given equation using radians
Re-write the given equation in terms of radians:
[tex]\[ \frac{\int (x - 10) \, dx}{\frac{\pi}{2}} = \frac{\pi}{9} \][/tex]
Step 4: Solve the integral
Integrate the function [tex]\( x - 10 \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \int (x - 10) \, dx = \frac{x^2}{2} - 10x \][/tex]
Step 5: Plug the integral result into the equation
Rewrite the equation with the integral result:
[tex]\[ \frac{\frac{x^2}{2} - 10x}{\frac{\pi}{2}} = \frac{\pi}{9} \][/tex]
Simplify the fraction on the left:
[tex]\[ \left(\frac{x^2}{2} - 10x \right) \div \frac{\pi}{2} = \left(\frac{x^2}{2} - 10x \right) \times \frac{2}{\pi} \][/tex]
[tex]\[ \frac{2}{\pi} \left(\frac{x^2}{2} - 10x\right) = \frac{\pi}{9} \][/tex]
[tex]\[ \frac{x^2}{\pi} - \frac{20x}{\pi} = \frac{\pi}{9} \][/tex]
Step 6: Clear the fraction by multiplying both sides by [tex]\(\pi\)[/tex]
Multiply both sides by [tex]\(\pi\)[/tex] to clear the fraction:
[tex]\[ x^2 - 20x = \frac{\pi^2}{9} \][/tex]
Step 7: Solve the quadratic equation
Rearrange the equation into standard quadratic form:
[tex]\[ x^2 - 20x - \frac{\pi^2}{9} = 0 \][/tex]
To solve the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here [tex]\( a = 1 \)[/tex], [tex]\( b = -20 \)[/tex], and [tex]\( c = -\frac{\pi^2}{9} \)[/tex].
Compute the discriminant:
[tex]\[ b^2 - 4ac = (-20)^2 - 4 \cdot 1 \cdot \left(-\frac{\pi^2}{9} \right) \][/tex]
[tex]\[ = 400 + \frac{4\pi^2}{9} \][/tex]
[tex]\[ = \frac{3600 + 4\pi^2}{9} \][/tex]
Then, compute the roots:
[tex]\[ x = \frac{20 \pm \sqrt{\frac{3600 + 4\pi^2}{9}}}{2} \][/tex]
[tex]\[ x = \frac{20 \pm \frac{\sqrt{3600 + 4\pi^2}}{3}}{2} \][/tex]
[tex]\[ x = \frac{20 \pm \frac{2\sqrt{900 + \pi^2}}{3}}{2} \][/tex]
[tex]\[ x = 10 \pm \frac{\sqrt{900 + \pi^2}}{3} \][/tex]
Therefore, the solutions are:
[tex]\[ x = 10 + \frac{\sqrt{900 + \pi^2}}{3} \][/tex]
[tex]\[ x = 10 - \frac{\sqrt{900 + \pi^2}}{3} \][/tex]
Numerically, these approximate values yield two solutions:
[tex]\[ x \approx 22.7606 \][/tex]
[tex]\[ x \approx -2.7606 \][/tex]
Thus, the two solutions to the equation are:
[tex]\[ x \approx 22.7606 \][/tex]
[tex]\[ x \approx -2.7606 \][/tex]
Step 1: Understand the given equation
The equation provided is:
[tex]\[ \frac{\int (x - 10) \, dx}{90^\circ} = 20^\circ \][/tex]
Step 2: Convert degrees to radians
We need to work with radians instead of degrees for the integration. The conversion factor between degrees and radians is:
[tex]\[ 1^\circ = \frac{\pi}{180} \text{ radians} \][/tex]
So, we convert [tex]\( 20^\circ \)[/tex] to radians:
[tex]\[ 20^\circ = 20 \times \frac{\pi}{180} = \frac{20\pi}{180} = \frac{\pi}{9} \text{ radians} \][/tex]
Similarly, [tex]\( 90^\circ \)[/tex] in radians is:
[tex]\[ 90^\circ = 90 \times \frac{\pi}{180} = \frac{90\pi}{180} = \frac{\pi}{2} \text{ radians} \][/tex]
Step 3: Re-write the given equation using radians
Re-write the given equation in terms of radians:
[tex]\[ \frac{\int (x - 10) \, dx}{\frac{\pi}{2}} = \frac{\pi}{9} \][/tex]
Step 4: Solve the integral
Integrate the function [tex]\( x - 10 \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \int (x - 10) \, dx = \frac{x^2}{2} - 10x \][/tex]
Step 5: Plug the integral result into the equation
Rewrite the equation with the integral result:
[tex]\[ \frac{\frac{x^2}{2} - 10x}{\frac{\pi}{2}} = \frac{\pi}{9} \][/tex]
Simplify the fraction on the left:
[tex]\[ \left(\frac{x^2}{2} - 10x \right) \div \frac{\pi}{2} = \left(\frac{x^2}{2} - 10x \right) \times \frac{2}{\pi} \][/tex]
[tex]\[ \frac{2}{\pi} \left(\frac{x^2}{2} - 10x\right) = \frac{\pi}{9} \][/tex]
[tex]\[ \frac{x^2}{\pi} - \frac{20x}{\pi} = \frac{\pi}{9} \][/tex]
Step 6: Clear the fraction by multiplying both sides by [tex]\(\pi\)[/tex]
Multiply both sides by [tex]\(\pi\)[/tex] to clear the fraction:
[tex]\[ x^2 - 20x = \frac{\pi^2}{9} \][/tex]
Step 7: Solve the quadratic equation
Rearrange the equation into standard quadratic form:
[tex]\[ x^2 - 20x - \frac{\pi^2}{9} = 0 \][/tex]
To solve the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here [tex]\( a = 1 \)[/tex], [tex]\( b = -20 \)[/tex], and [tex]\( c = -\frac{\pi^2}{9} \)[/tex].
Compute the discriminant:
[tex]\[ b^2 - 4ac = (-20)^2 - 4 \cdot 1 \cdot \left(-\frac{\pi^2}{9} \right) \][/tex]
[tex]\[ = 400 + \frac{4\pi^2}{9} \][/tex]
[tex]\[ = \frac{3600 + 4\pi^2}{9} \][/tex]
Then, compute the roots:
[tex]\[ x = \frac{20 \pm \sqrt{\frac{3600 + 4\pi^2}{9}}}{2} \][/tex]
[tex]\[ x = \frac{20 \pm \frac{\sqrt{3600 + 4\pi^2}}{3}}{2} \][/tex]
[tex]\[ x = \frac{20 \pm \frac{2\sqrt{900 + \pi^2}}{3}}{2} \][/tex]
[tex]\[ x = 10 \pm \frac{\sqrt{900 + \pi^2}}{3} \][/tex]
Therefore, the solutions are:
[tex]\[ x = 10 + \frac{\sqrt{900 + \pi^2}}{3} \][/tex]
[tex]\[ x = 10 - \frac{\sqrt{900 + \pi^2}}{3} \][/tex]
Numerically, these approximate values yield two solutions:
[tex]\[ x \approx 22.7606 \][/tex]
[tex]\[ x \approx -2.7606 \][/tex]
Thus, the two solutions to the equation are:
[tex]\[ x \approx 22.7606 \][/tex]
[tex]\[ x \approx -2.7606 \][/tex]
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