a) In the diagram, $PQ$ is parallel to $BC$. $APB$ and $AQC$ are straight lines. If $PQ= 8$ cm, $BC = 10$ cm and $AB = 9$ cm. Calculate $PB$. b) Two glasses are mathematically simi... a) In the diagram, $PQ$ is parallel to $BC$. $APB$ and $AQC$ are straight lines. If $PQ= 8$ cm, $BC = 10$ cm and $AB = 9$ cm. Calculate $PB$. b) Two glasses are mathematically similar. The larger glass has a capacity of 0.5 litres and the smaller glass has a capacity of 0.25 litres. If the height of the larger glass is 13 cm. Calculate the height of the smaller glass. Read more

Steps to Solve

1. Set up similar triangles for part (a)

Since $PQ$ is parallel to $BC$, triangle $APQ$ is similar to triangle $ABC$. Therefore, the ratios of corresponding sides are equal:

$$ \frac{AP}{AB} = \frac{PQ}{BC} $$

2. Substitute given values

Substitute the given values $PQ = 8$ cm, $BC = 10$ cm, and $AB = 9$ cm into the equation:

$$ \frac{AP}{9} = \frac{8}{10} $$

3. Solve for $AP$

Multiply both sides of the equation by 9 to solve for $AP$:

$$ AP = \frac{8}{10} \times 9 = \frac{72}{10} = 7.2 \text{ cm} $$

4. Calculate $PB$

We know that $AB = AP + PB$. Therefore, $PB = AB - AP$. Substitute the values of $AB$ and $AP$:

$$ PB = 9 - 7.2 = 1.8 \text{ cm} $$

5. Find the volume scale factor for part (b)

The volume scale factor is the ratio of the volumes of the two glasses:

$$ \text{Volume Scale Factor} = \frac{0.5}{0.25} = 2 $$

6. Find the linear scale factor Since the glasses are mathematically similar, the ratio of their volumes is the cube of the ratio of their corresponding lengths (heights in this case). Therefore, if $k$ is the linear scale factor, then $k^3$ is the volume scale factor:

$$ k^3 = 2 $$ $$ k = \sqrt[3]{2} $$

7. Solve for height of smaller glass Let $h$ be the height of the smaller glass. Then, since the larger glass has a height of 13cm, we can write the equation:

$$ \frac{13}{h} = \sqrt[3]{2} $$

Solving for $h$:

$$ h = \frac{13}{\sqrt[3]{2}} $$

To rationalize the denominator, we can approximate $\sqrt[3]{2} \approx 1.2599$

$$ h \approx \frac{13}{1.2599} \approx 10.32 \text{cm} $$

Alternatively, let $h$ be the height of the smaller glass. Then $$ \frac{h}{13} = \frac{1}{\sqrt[3]{2}} $$ $$ h = \frac{13}{\sqrt[3]{2}} \approx 10.32 \text{cm} $$