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# math problem

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The equation of line $l$ is $y = -x + 6$.  The equation of line $m$ is $y = -x + 1$.  What is the probability that a point randomly selected in the first quadrant and below $l$ will fall between $l$ and $m$? Express your answer as a decimal to the nearest hundredth.

Aug 9, 2023

#1
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Below is an image that representst this situation well. We can find the area of the triangle created by $$y = -x + 6$$. This is just the area of a right triangle. $$A_{\triangle 1}= \frac{1}{2} * 6 * 6 = 18$$. We can then find the area underneath the equation $$y = -x + 1$$. In this case, $$A_{\triangle 2} = \frac{1}{2} * 1 * 1 = \frac{1}{2}$$.

The probability that a point lies between $$y = -x + 6$$ and $$y = -x + 1$$ is essentially finding the ratio of the part and the whole. The whole is the area underneath the triangle created by y = -x + 6 and the boundaries of the first quadrant. The part is the area between y = -x + 6 and y = -x + 1 in the first quadrant.

$$\frac{\text{part}}{\text{whole}} = \frac{18 - \frac{1}{2}}{18} = \frac{35}{36} \approx 0.97$$. This is rounded to the nearest hundreds place, as suggested.

Aug 9, 2023

#1
-1

Below is an image that representst this situation well. We can find the area of the triangle created by $$y = -x + 6$$. This is just the area of a right triangle. $$A_{\triangle 1}= \frac{1}{2} * 6 * 6 = 18$$. We can then find the area underneath the equation $$y = -x + 1$$. In this case, $$A_{\triangle 2} = \frac{1}{2} * 1 * 1 = \frac{1}{2}$$.

The probability that a point lies between $$y = -x + 6$$ and $$y = -x + 1$$ is essentially finding the ratio of the part and the whole. The whole is the area underneath the triangle created by y = -x + 6 and the boundaries of the first quadrant. The part is the area between y = -x + 6 and y = -x + 1 in the first quadrant.

$$\frac{\text{part}}{\text{whole}} = \frac{18 - \frac{1}{2}}{18} = \frac{35}{36} \approx 0.97$$. This is rounded to the nearest hundreds place, as suggested.

The3Mathketeers Aug 9, 2023