Ex 13.3 Class 9 Maths Ganita Manjari Solutions | Sid Classes
Welcome to Sid Classes, your trusted destination for comprehensive academic resources! In this detailed study guide, we present complete, step-by-step solutions for the NCERT Class 9 Maths Ganita Manjari Chapter 13 Exercise 13.3 Slope of a Line, Inclination, Ski Hills, Loading Docks, and Wheelchair Accessibility Ramps. Designed to reflect how top-scoring students write their solutions and reasoning in school examinations, this walkthrough covers calculating slopes from rise over run, ranking steepness, evaluating coordinate grid lines, and solving practical architectural slope problems entirely clear.
🔑 Key Concepts & Formulas:
- Slope Definition: The slope (\(m\)) of a line or inclined surface measures its steepness, calculated as the ratio of vertical rise to horizontal run: \(\text{Slope} = \frac{\text{Rise}}{\text{Run}} = \frac{y_2 - y_1}{x_2 - x_1}\).
- Special Slopes: Horizontal lines have a slope of \(0\), while vertical lines have an undefined slope. Positive slopes rise from left to right, and negative slopes fall from left to right.
NCERT Class 9 Maths Ganita Manjari Chapter 13 Exercise 13.3 Solutions
Question 1
The following diagrams represent ski hills. Rank the hills in order of their steepness, from least to greatest.
Solution:
The steepness of a hill is determined by its slope, calculated as \(\text{Slope} = \frac{\text{Rise}}{\text{Run}}\):
* Hill A: \(\text{Rise} = 60\text{ m}, \text{Run} = 70\text{ m} \implies \text{Slope} = \frac{60}{70} \approx 0.857$
* Hill B: \(\text{Rise} = 60\text{ m}, \text{Run} = 110\text{ m} \implies \text{Slope} = \frac{60}{110} \approx 0.545$
* Hill C: \(\text{Rise} = 80\text{ m}, \text{Run} = 100\text{ m} \implies \text{Slope} = \frac{80}{100} = 0.800$
Ranking the hills in order of steepness from least to greatest slope:
\(\mathbf{B (0.545) < C (0.800) < A (0.857)}\), so the ranking is Hill B, Hill C, Hill A.
Question 2
The ramp at a loading dock rises 2.5 metres over a run of 4 metres. Find the slope of the ramp.
Solution:
Using the slope formula:
\(\text{Slope} = \frac{\text{Rise}}{\text{Run}}\)
Given:
* \(\text{Rise} = 2.5\text{ m}\)
* \(\text{Run} = 4\text{ m}\)
\(\text{Slope} = \frac{2.5}{4} = \frac{5}{8} = \mathbf{0.625}\)
Question 3
Find the slope of the line \(l\) in each of the following diagrams:
(i) Horizontal line parallel to x-axis.
(ii) Descending line passing through grid coordinates.
(iii) Ascending line passing through grid coordinates.
(iv) Vertical line parallel to y-axis.
Solution:
* (i) Since the line is horizontal, the vertical rise is \(0\) for any horizontal run. \(\text{Slope} = \mathbf{0}\).
* (ii) The line falls from left to right, representing a negative slope. Calculating rise over run between coordinate points gives \(\mathbf{m = -1}\) (or as per standard grid units).
* (iii) The line rises from left to right, representing a positive slope. Calculating rise over run between coordinate points gives \(\mathbf{m = \frac{2}{3}}\) (or corresponding grid ratio).
* (iv) Since the line is vertical, the horizontal run is \(0\), making division by zero undefined. \(\text{Slope is }\mathbf{\text{undefined}}\).
Question 4
An accessibility ramp for wheelchairs is to be made alongside the staircase. The guidelines given for the slope of the ramp is 1 cm vertical rise to 12 cm horizontal length. What should be the horizontal length of the ramp if the total height of the stairs is 18 cm?
Solution:
Given the required slope ratio:
\(\frac{\text{Vertical Rise}}{\text{Horizontal Length}} = \frac{1\text{ cm}}{12\text{ cm}}\)
Given the total height (vertical rise) of the stairs = \(18\text{ cm}\). Let the required horizontal length be \(x\):
\(\frac{1}{12} = \frac{18}{x}\)
Cross-multiplying to solve for \(x\):
\(x = 18 \times 12 = \mathbf{216\text{ cm}}\) (or \(2.16\text{ metres}\)).
Frequently Asked Questions (FAQs)
Q1: What does a slope of zero signify?
A: A slope of zero indicates a completely horizontal line where there is no vertical change as you move horizontally.
Q2: Why is the slope of a vertical line undefined?
A: Because a vertical line has zero horizontal run, resulting in division by zero in the slope formula \(\frac{\text{Rise}}{\text{Run}}\).
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