Question

1(a) A number is written as 37 in base.Twice the number is written as 75 in base x.Find the value of x
(B)A garden ,in front of a square is surrounded by a path which is 1m wide on two opposite sides and 1/2m wide on the two opposite sides.If the area of the path is 17m^2,

calculate the :
1.perimeter of the garden
2.area of the space covered by the garden and the path

Answer 1

Answer on Question #64301 – Math – Algebra

Question

1(a) A number is written as 37 in base x. Twice the number is written as 75 in base x. Find the value of x

Solution

1 (a) If $x$ is the base, then $(3x + 7) * 2 = 7x + 5 \rightarrow x = 9$.

Answer: $x = 9$.

Question

(B) A garden, in front of a square is surrounded by a path which is 1m wide on two opposite sides and 1/2m wide on the two opposite sides. If the area of the path is $17\mathrm{m}^2$,

calculate the :

1. perimeter of the garden

2. area of the space covered by the garden and the path

Solution

(B) Let $x$ be the side of the garden.

Area of the path:

$$\begin{array}{l} (x + 1 + 1)(x + 0.5 + 0.5) - x^2 = 17 \rightarrow (x + 2)(x + 1) - x^2 = 17 \rightarrow \\ \rightarrow x^2 + 3x + 2 - x^2 = 17 \rightarrow 3x + 2 = 17 \rightarrow 3x = 15 \rightarrow x = 5 \text{ m}. \end{array}$$

1. Perimeter:

$$P = 4x = 4 \cdot 5 = 20 \text{ m}.$$

2. Area of the garden:

$$S_g = x^2 = 5^2 = 25 \text{ m}^2.$$

Area of the path:

$$S_p = 17 \text{ m}^2.$$

Area of the space covered by the garden and the path:

$$\begin{array}{l} S = S_g + S_p = 25 + 17 = 42 \text{ m}^2 \text{ or} \\ S = (x + 2)(x + 1) = (5 + 2) \cdot (5 + 1) = 7 \cdot 6 = 42 \text{ m}^2. \end{array}$$

Answer: 1. 20 m. 2. 42 m².

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