Theory question 111
Ratio, Percentages and Rates(a) If (3p + 4q)/(3p − 4q) = 2, find p : q
(b) The diagram shows the cross section of a bridge with a semi circular hollow in the middle. If the perimeter of the cross section is 34 m, calculate the: (i) length PQ; (ii) area of the cross section [Take π = 22/7]
Show the model answer
Model answer
(a) 3p + 4q = 6p − 8q → 3p = 12q → p : q = 4 : 1
(b)(i) Let |PQ| = L; diameter of semicircle TS = L − 4, radius (L − 4)/2
Perimeter = L + 4 + 2 + 2 + 4 + π(L − 4)/2 = 34
L + 11(L − 4)/7 = 22 → 18L = 198 → |PQ| = 11 m
(ii) Radius = 3.5 m; area = 11 × 4 − ½ × 22/7 × 3.5² = 44 − 19.25 = 24.75 m²
Explanation
The perimeter includes the semicircular arc, not the straight gap TS.