Mathematics 10 — Exercise 7.3 · Q2

Full question

A pyramid has rectangular base 16×10 cm; its vertex is 15 cm vertically above the centre M. Find the angle of the vertex-to-P line with the base.

Step-by-step solution

1. Calculate this operation before combining the remaining terms.

   $${16}^2=256$$

2. Calculate this operation before combining the remaining terms.

   $${10}^2=100$$

3. Calculate this operation before combining the remaining terms.

   $$256+100=356$$

4. Extract a perfect-square factor from under the root.

   $$356=4\times 89$$

5. Use √(ab)=√a×√b for nonnegative a,b; take the root of the perfect square.

   $$\sqrt{356}=2\times \sqrt{89}$$

6. Find the base diagonal by Pythagoras.

   $$\sqrt{{16}^2+{10}^2}$$

7. Square both dimensions and add.

   $$256+100$$

8. Take the positive square root for the complete diagonal.

   $$18.867962$$

9. M is the centre, so PM is half this diagonal. Divide by 2.

   $$\mathrm{PM}=\frac{18.867962}{2}$$

10. The vertical height is opposite and PM adjacent to the required angle.

    $$\tan \left(\mathrm{\theta °}\right)=\frac{15}{9.4339811}$$

11. Evaluate the quotient, then use inverse tangent.

    $$57.832896$$

12. Round to one decimal place in degrees.

Answer

The final answer is shown below.