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Gabriele R.

Maths tutor for over 10 years

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Portuguese

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Question:

Complete as seguintes frases: 1 - Pelé foi ______________ o melhor jogador de futebol do século XX. (a) com certeza (b) concerteza (c) com certesa 2 - Ela sempre resolve os problemas com bastante ______________ (a) descrição (b) discrição (c) ambas se aplicam 3 - Camila falou que queria ____ exemplos, ____ não falou quantos. (a) mais - mais (b) mais - mas (c) mas - mais

Gabriele R.

Answer:

1 - a 2 - b 3 - b

Trigonometry

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Question:

The lengths of side $$AB$$ and side $$BC$$ of a scalene triangle $$ABC$$ are 12 cm and 9 cm respectively. The size of angle $$C$$ is 57 $$^{\circ}$$. Find the length of side $$AC$$.

Gabriele R.

Answer:

Let $$x$$ be the length of side $$AC$$. We already know two sides of the triangle and one of its angles. According to the law of cosines, $$AB^2 = x^2 + AC^2 - 2 \cdot x \cdot AC \cdot \cos(C)$$ $$12^2 = x^2 + 9^2 - 2 \cdot x \cdot 9 \cdot \cos(57^{\circ})$$ $$144 = x^2 + 81 - 2 \cdot x \cdot 9 \cdot 0.5446$$ $$x^2 -9.8x -63=0$$ Solving the quadratic equation for $$x$$: $$x = 14$$ and $$x = -4$$ $$x$$ cannot be negative as it's a side of a triangle. Therefore the solution is $$x = 14$$ (rounded to one decimal place)

Algebra

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Question:

A fuel station sells 10,000 liters of diesel per day at $1.50/ liter. Its owner realized that for each cent of discount he gives per liter, the sales increase in 100 liters. Considering $$x$$ as the value (in cents) of the discount given in the price of each liter of diesel , and $$R$$ the total income, model $$R$$ as a function of $$x$$.

Gabriele R.

Answer:

The revenue $$R$$ - total income per day - is calculated by the product between the quantity of liters sold, $$Q$$, for each liter's price, $$P$$. These two variables are a function of the discount value $$x$$, in cents. $$R = Q\cdot P$$ $$(1)$$ The sale is 10,000 liters per day at the price of $1.50/ liter. At each cent of the discount, the quantity sold increases by 100 liters, so, for example, $0.01 discount increases 100 liters, $0.02 200 liters and so on. According to that we can express $$Q$$ and $$P$$ in terms of $$x$$, as follows: $$Q = 10,000 + 100\cdot x$$ $$(2)$$ $$P = 1.50 - 0.01\cdot x$$ $$(3)$$ Multiplying $$(2)$$ and $$(3)$$ $$R = Q\cdot P$$ $$R = (10,000 + 100x)\cdot (1.50 - 0.01 x) $$ $$R = 10,000\cdot 1.50 - (10,000\cdot0.01x) + 100x\cdot 1.50 - (100x\cdot 0.01x) $$ $$\therefore$$ $$R(x) = 15.000 + 50x – x^2 $$

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