Mathematics 10 — Exercise 7.3 · Q5

Full question

A prism has rectangular base AB=10 cm, BC=7 cm. M is the midpoint of AD and TM=6 cm is vertical. Find the angle of TB with the base.

Step-by-step solution

1. AD=BC=7 cm and M bisects AD, so AM=7/2=3.5 cm.

2. Calculate this operation before combining the remaining terms.

   $${10}^2=100$$

3. For multiplication of fractions, multiply the numerators together and the denominators together.

   $$\frac{7\times 1}{1\times 2}$$

4. Calculate these two products separately.

   $$\frac{7}{2}$$

5. Calculate this operation before combining the remaining terms.

   $$7\times \frac{1}{2}=\frac{7}{2}$$

6. Calculate this operation before combining the remaining terms.

   $${\left(\frac{7}{2}\right)}^2=\frac{49}{4}$$

7. To add these fractions, first use a common denominator: the least common multiple of their denominators.

   $$\mathrm{LCD}=4$$

8. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

   $$100\times \frac{1}{1}=\frac{100\times 4}{1\times 4}$$

9. Calculate the converted numerator; the denominator is now common.

   $$\frac{400}{4}$$

10. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{49}{4}=\frac{49\times 1}{4\times 1}$$

11. Calculate the converted numerator; the denominator is now common.

    $$\frac{49}{4}$$

12. Keep the common denominator and add the signed numerators.

    $$\frac{400+49}{4}=\frac{449}{4}$$

13. Calculate this operation before combining the remaining terms.

    $$100+\frac{49}{4}=\frac{449}{4}$$

14. In the base right triangle AMB use MB²=10²+3.5².

    $${10}^2+{\left(\frac{7}{2}\right)}^2$$

15. Square and add: 100+12.25=112.25, then take the square root.

    $$10.59481$$

16. Calculate this operation before combining the remaining terms.

    $$\frac{1}{\frac{1059481}{100000}}=\frac{100000}{1059481}$$

17. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{6\times 100000}{1\times 1059481}$$

18. Calculate these two products separately.

    $$\frac{600000}{1059481}$$

19. Calculate this operation before combining the remaining terms.

    $$6\times \frac{100000}{1059481}=\frac{600000}{1059481}$$

20. TM is vertical, so the angle between TB and the base is angle TBM. Use tan θ=TM/MB.

    $$\frac{6}{10.59481}$$

21. Calculate the quotient and take inverse tangent.

    $$29.523529$$

Answer

The final answer is shown below.