Question 8/20: Set 6 - Try Out - Mathematical Reasoning (SNBT)

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Number 8

A construction worker wants to cut a wooden plank into $6$ pieces. The length of each piece follows a geometric series. If the shortest piece is $3$ cm and the longest is $96$ cm.

Then the total length of the original wooden plank is ....

$175$ cm

$189$ cm

$192$ cm

$221$ cm

$235$ cm

Explanation

Determining the Ratio

It is given that the wood is cut into 6 pieces, so $n = 6$ . The lengths of the pieces form a geometric sequence.

We assume the first term is the longest (it works the same way if assumed otherwise for the total sum). Let the first term be $a = 96$ and the 6th term be the shortest $U_6 = 3$ .

The formula for the $n$ -th term of a geometric sequence is:

U_n = ar^{n-1}

Substitute the known values:

U_6 = 96 \cdot r^{6-1}

3 = 96 \cdot r^5

Simplify the equation to find the ratio ( $r$ ):

r^5 = \frac{3}{96}

r^5 = \frac{1}{32}

r^5 = \left(\frac{1}{2}\right)^5

r = \frac{1}{2}

So, the ratio between the wood pieces is $\frac{1}{2}$ .

Calculating Total Wood Length

The total length of the original wood is the sum of all pieces ( $S_6$ ). Since $r < 1$ , we use the formula:

S_n = \frac{a(1 - r^n)}{1 - r}

Substitute $a = 96$ , $r = \frac{1}{2}$ , and $n = 6$ :

S_6 = \frac{96\left(1 - \left(\frac{1}{2}\right)^6\right)}{1 - \frac{1}{2}}

S_6 = \frac{96\left(1 - \frac{1}{64}\right)}{\frac{1}{2}}

Calculate the result:

S_6 = 96 \cdot \left(\frac{63}{64}\right) \cdot 2

S_6 = 192 \cdot \frac{63}{64}

S_6 = 3 \cdot 63

S_6 = 189

Thus, the total length of the original wooden plank is $189$ cm.

Command Palette

Number 8

Explanation

Determining the Ratio

Calculating Total Wood Length