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Grace is putting some stairs into a new house - Junior Cycle Mathematics - Question 10 - 2018
Question 10
Grace is putting some stairs into a new house. The diagram on the right shows three of the steps in the stairs. Each step consists of a riser and a tread. All of the... show full transcript
Worked Solution & Example Answer:Grace is putting some stairs into a new house - Junior Cycle Mathematics - Question 10 - 2018
Step 1
Grace's stairs will climb a total vertical height of 2.1 m (i.e. the sum of the risers is 2.1 m). Work out the horizontal length of Grace's stairs (i.e. the sum of the treads).
Answer
To find the total horizontal length of Grace's stairs, we first need to determine the total number of risers. Given that the height of each riser in a private building is 175 mm, we can calculate the number of risers as follows:
- Total vertical height of stairs = 2.1 m = 2100 mm.
- Number of risers = Total height / Height of each riser = ( \frac{2100 \text{ mm}}{175 \text{ mm}} = 12 \text{ risers} ).
Next, we calculate the total horizontal length (sum of the treads):
- Each tread in a private building is 250 mm.
- Total horizontal length = Number of treads × Length of each tread = 11 × 250 mm = 2750 mm.
Step 2
The angle of elevation of Grace's stairs is marked A in the diagram on the right. Using trigonometry, and without measuring, work out the size of the angle A. Give your answer correct to the nearest degree.
Answer
To determine angle A, we can use the tangent function from trigonometry:
- ( \tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{total height}}{\text{total horizontal length}} = \frac{2100 \text{ mm}}{2750 \text{ mm}} ).
- Calculating the tangent: ( \tan(A) = 0.7636 ).
- Using the arctangent function: ( A = \arctan(0.7636) \approx 37.4^{\circ} ).
Thus, rounding to the nearest degree, angle A is approximately 37°.
Step 3