(a) If \(\frac{3p + 4q}{3p - 4q} = 2\), find \(p : q\). (b) The diagram shows the cross section of a bridge with a semi-circular hollow in the middle. If th...

Answer Details

(a) Find \(p:q\).

\[\frac{3p+4q}{3p-4q}=2\]

Cross multiply:

\[3p+4q=2(3p-4q)=6p-8q\]\[3p+4q-6p+8q=0\]\[-3p+12q=0\Rightarrow 3p=12q\Rightarrow p=4q\]\[p:q=\mathbf{4:1}\]

(b) The bridge cross-section.

From the diagram, \(PQRU\) is a rectangle with the two vertical sides \(PU=QR=4\text{ m}\), and a semicircular hollow is cut from the base. Along the base the two flat pieces are \(UT=SR=2\text{ m}\), and \(TS\) is the diameter of the semicircle. Let the radius be \(r\), so \(TS=2r\).

(i) Length \(PQ\).

The outline (perimeter) is the top \(PQ\), the two sides, the two flat base pieces, and the semicircular arc:

\[\text{Perimeter}=PQ+PU+QR+UT+SR+\text{arc}(TS)\]

Since \(PQRU\) is a rectangle, \(PQ=UR=UT+TS+SR=2+2r+2=2r+4\). With arc \(=\pi r\):

\[34=(2r+4)+4+4+2+2+\pi r\]\[34=2r+16+\pi r\]\[18=r\left(2+\tfrac{22}{7}\right)=r\cdot\tfrac{36}{7}\]\[r=\frac{18\times7}{36}=3.5\text{ cm}\]

Therefore

\[PQ=2r+4=2(3.5)+4=\mathbf{11\text{ cm}}\]

(ii) Area of the cross-section.

Area \(=\) rectangle \(-\) semicircle:

\[=PQ\times4-\tfrac{1}{2}\pi r^{2}=11\times4-\tfrac{1}{2}\cdot\tfrac{22}{7}\cdot(3.5)^{2}\]\[=44-\tfrac{1}{2}\cdot\tfrac{22}{7}\cdot12.25=44-19.25=\mathbf{24.75\text{ cm}^{2}}\]

(The heights and base pieces read \(4\) and \(2\) from the figure; with the stated perimeter \(34\) this gives a consistent \(r=3.5\).)