Question: User: A
1)
a
46 m
30 m
Calculate length a to
1 decimal place.
2)
30 m
10 m
b
Calculate the size of angle
b to 2 decimal places.
3)
25 m
55°
c
Calculate length c to
1 decimal place.
B
1)
17 mm
d cm
5.2 cm
Calculate d to
3 significant figures.
2)
24 cm
35 cm
e
Calculate angle e to
1 decimal place.
3)
12 m
f
19°
Calculate length f to
1 decimal place.
C
1)
This is a
square-based prism.
A
B
10 cm
5 cm
Express length AB in the form 2a2 where a is
an integer.
2)
58°
24 m
16 m
x6
Calculate the value of
x to 3 decimal places.
1. **Problem:** Calculate length $a$ in a right triangle with base $46$ m and vertical side $30$ m.
2. **Formula:** Use Pythagoras theorem: $$a = \sqrt{46^2 + 30^2}$$
3. **Calculation:**
$$a = \sqrt{2116 + 900} = \sqrt{3016}$$
4. **Simplify:**
$$a \approx 54.922$$
5. **Answer:** Rounded to 1 decimal place:
$$a = 54.9 \text{ m}$$
---
1. **Problem:** Calculate angle $b$ in a right triangle with hypotenuse $30$ m and vertical side $10$ m.
2. **Formula:** Use sine function:
$$\sin b = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{10}{30} = \frac{1}{3}$$
3. **Calculate angle:**
$$b = \sin^{-1}\left(\frac{1}{3}\right) \approx 19.471^\circ$$
4. **Answer:** Rounded to 2 decimal places:
$$b = 19.47^\circ$$
---
1. **Problem:** Calculate length $c$ in a right triangle with hypotenuse $25$ m and angle $55^\circ$ opposite side $c$.
2. **Formula:** Use sine function:
$$\sin 55^\circ = \frac{c}{25} \Rightarrow c = 25 \times \sin 55^\circ$$
3. **Calculate:**
$$c = 25 \times 0.819152 = 20.4788$$
4. **Answer:** Rounded to 1 decimal place:
$$c = 20.5 \text{ m}$$
---
1. **Problem:** Calculate length $d$ in a right triangle with sides $17$ mm and $5.2$ cm (convert units), angle at top.
2. **Convert units:**
$$17 \text{ mm} = 1.7 \text{ cm}$$
3. **Formula:** Use Pythagoras theorem:
$$d = \sqrt{1.7^2 + 5.2^2}$$
4. **Calculate:**
$$d = \sqrt{2.89 + 27.04} = \sqrt{29.93}$$
5. **Answer:** Rounded to 3 significant figures:
$$d = 5.47 \text{ cm}$$
---
1. **Problem:** Calculate angle $e$ in a right triangle with sides $24$ cm and $35$ cm, right angle at top.
2. **Formula:** Use tangent function:
$$\tan e = \frac{24}{35}$$
3. **Calculate:**
$$e = \tan^{-1}\left(\frac{24}{35}\right) \approx 34.49^\circ$$
4. **Answer:** Rounded to 1 decimal place:
$$e = 34.5^\circ$$
---
1. **Problem:** Calculate length $f$ in a right triangle with side $12$ m adjacent to angle $19^\circ$, hypotenuse $f$.
2. **Formula:** Use cosine function:
$$\cos 19^\circ = \frac{12}{f} \Rightarrow f = \frac{12}{\cos 19^\circ}$$
3. **Calculate:**
$$f = \frac{12}{0.945518} = 12.69$$
4. **Answer:** Rounded to 1 decimal place:
$$f = 12.7 \text{ m}$$
---
1. **Problem:** Express length $AB$ in a square-based prism with base edges $10$ cm and vertical edge $5$ cm, $AB$ is the diagonal from front-bottom to back-top.
2. **Formula:** Use 3D distance formula:
$$AB = \sqrt{10^2 + 10^2 + 5^2} = \sqrt{100 + 100 + 25} = \sqrt{225}$$
3. **Answer:**
$$AB = \sqrt{225} = 15 \text{ cm}$$
Expressed as $\sqrt{a}$:
$$AB = \sqrt{225}$$
---
1. **Problem:** Calculate angle $x$ in a triangle with angles $58^\circ$, $x^\circ$, and right angle $90^\circ$.
2. **Formula:** Sum of angles in triangle is $180^\circ$:
$$x = 180 - 90 - 58 = 32^\circ$$
3. **Answer:** Rounded to 3 decimal places:
$$x = 32.000^\circ$$