# Tutor profile: Karishma S.

## Questions

### Subject: Trigonometry

Prove the trigonometric identity: 4cos(pi/6-x)sin(pi/3-x) = sin3x/sin x If cot(x) = 2 then find (2 + 2sinx)(1 - sin x) / (1 + cos x)(2 - 2cosx)

cot(x) = 2(given) (2 + 2sinx)(1 - sin x) / (1 + cos x)(2 - 2cosx) = (2 + 2sinx)(1 - sin x) / (2 - 2cosx)(1 + cos x) = 2(1 + sinx)(1 - sin x) / 2(1 - cosx)(1 + cos x) ( //taking 2 common) = 1 - sin^2 x / 1 - cos^2 x (// (a+b)(a-b) = a^2- b^2) = cos^2 x / sin^2 x (// from identity sin^2 x + cos^2 x = 1) = cot^2 x (// cos x / sin x = cot x) = 2^2 = 4

### Subject: Geometry

Q.How many cubic cm of wood are there in a box,which measures 24cm by 22cm by 17cm,thickness of the wood is 1cm.

By considering the dimensions of Box, we have Length - 24 cm, Breadth - 22 cm and Height -17 cm. So, External Volume of Box = LxBxH = 24x22x17= 8976 cm^3 Thickness of the wood= 1 cm => Internal Length of box = 24-2* Thickness= 24-2= 22 cm Internal Breadth of box = 22-2* Thickness= 22-2= 20 cm Internal Height of box = 17-2* Thickness= 17-2= 15 cm => Internal volume of box= 22x20x15= 6600 cm^3 Thus, The volume of wood = External Volume - Internal Volume = 8976 - 6600 = 2376 cm^3

### Subject: Basic Math

The manager of a weaving factory estimates that if 10 machines run at 100% efficiency for 8 hours, they will produce 1450 meters of cloth. Due to some technical problems, 4 machines run of 95% efficiency and the remaining 6 at 90% efficiency. How many meters of cloth can these machines will produce in 8 hours?

At 100% efficiency 1 machine produces 1450/10 = 145 m of cloth. At 95% efficiency, 4 machines produce 4 * 145 * 95/100 = 551 m of cloth. At 90% efficiency, 6 machines produce 6 * 145 * 90/100 = 783 m of cloth. Total cloth produced by all 10 machines = 551 + 783 = 1334 m

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