Question

(a) The diameter of a cylinder closed at both ends is 7cm. If the total surface area is 209cm^2, calculate the height. [Take pi=22/7]
(b) The points X and Y, 19m apart, are on the same side of a tree, on the same horizontal ground as the foot of the tree. The angles of elevation of the top, T, of the tree from X and Y are 20 degrees and 38 degrees respectively. Calculate the height of the tree.

Accepted Answer

(a) Radius r = 7/2 = 3.5cm. Total surface area (cylinder closed at both ends) = 2*pi*r*h + 2*pi*r^2 = 2*pi*r(h+r).
209 = 2 x (22/7) x 3.5 x (h+3.5) = 22 x (h+3.5).
h+3.5 = 209/22 = 9.5, so h = 9.5-3.5 = 6cm.

(b) Let F be the foot of the tree, height of tree = h, and let FY = a (Y being the nearer point, with the larger angle of elevation 38 degrees), so FX = a+19 (X the farther point, angle of elevation 20 degrees).
From triangle FTY: tan(38) = h/a, so a = h/tan(38).
From triangle FTX: tan(20) = h/(a+19), so a+19 = h/tan(20).
Subtracting: h/tan(20) - h/tan(38) = 19, i.e. h(cot20 - cot38) = 19.
cot(20) = 2.747, cot(38) = 1.280. h(2.747-1.280) = 19, so h(1.467) = 19, giving h = 19/1.467 = 12.95m (approx).
The height of the tree is approximately 12.95m (to 2 d.p.).

GCE Mathematics 2022 Theory — Question 13

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