Question
Question 6, 1.3.31 HW Score: 26.32\%, 5 of 19 points Points: 0 of 1 Suppose that the point \( (x, y) \) is in the indicated quadrant. Decide whether the given ratio is positive or negative. Recall that \( r=\sqrt{x^{2}+y^{2}} \). \[ 1, \frac{r}{y} \] In Quadrant I, is \( \frac{r}{y} \) positive or negative? Negative Positive
Ask by Parry Valdez. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
In Quadrant I, \( \frac{r}{y} \) is **positive**.
Solution
In **Quadrant I**, both \( x \) and \( y \) coordinates of a point \( (x, y) \) are **positive**. Given the ratio:
\[
\frac{r}{y}
\]
where \( r = \sqrt{x^{2} + y^{2}} \), let's analyze the sign of this ratio.
1. **\( r \) is always positive**:
\[
r = \sqrt{x^{2} + y^{2}} > 0 \quad \text{(since both \( x \) and \( y \) are positive in Quadrant I)}
\]
2. **\( y \) is positive in Quadrant I**:
\[
y > 0
\]
3. **The ratio \( \frac{r}{y} \)**:
\[
\frac{r}{y} = \frac{\text{Positive}}{\text{Positive}} = \text{Positive}
\]
Therefore, **\( \frac{r}{y} \) is positive in Quadrant I**.
**Answer:** Positive
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Simplify this solution Bonus Knowledge
In Quadrant I, both \( x \) and \( y \) are positive values. Since \( r \) is defined as \( r = \sqrt{x^2 + y^2} \), it will also be positive because it represents a distance from the origin. Thus, the ratio \( \frac{r}{y} \) is the quotient of two positive numbers, making it positive as well! If you're ever confused about signs in a given quadrant, remember the mnemonic "All Students Take Calculus," which helps you recall that in Quadrant I, all trigonometric functions (and by extension, their ratios) are positive!
