Solved on Feb 20, 2024

Find the length of the kite string if the kite's height is $12 \mathrm{m}$ and the string makes a $30^{\circ}$ angle with the ground.

STEP 1

Assumptions
1. The angle between the string of the kite and the ground is $30^{\circ}$ .
2. The height of the kite above the ground is $12 \mathrm{~m}$ .
3. The string is straight and forms a right-angled triangle with the ground and the height of the kite.
4. We are asked to find the length of the string, which is the hypotenuse of the right-angled triangle.

STEP 2

Draw a right-angled triangle to represent the situation. Label the height of the kite as the opposite side (O), the length of the string as the hypotenuse (H), and the angle between the string and the ground as $\theta = 30^{\circ}$ .

STEP 3

Use the trigonometric sine function, which relates the opposite side, the hypotenuse, and the angle in a right-angled triangle.
$\sin(\theta) = \frac{O}{H}$

STEP 4

Substitute the given values into the sine function.
$\sin(30^{\circ}) = \frac{12 \mathrm{~m}}{H}$

STEP 5

Calculate the value of $\sin(30^{\circ})$ .
$\sin(30^{\circ}) = \frac{1}{2}$

STEP 6

Substitute the value of $\sin(30^{\circ})$ into the equation.
$\frac{1}{2} = \frac{12 \mathrm{~m}}{H}$

STEP 7

Solve the equation for the hypotenuse (H) by multiplying both sides by H and then dividing by $\frac{1}{2}$ .
$H = \frac{12 \mathrm{~m}}{\frac{1}{2}}$

STEP 8

Calculate the length of the string (H).
$H = 12 \mathrm{~m} \times 2 = 24 \mathrm{~m}$
The length of the string that the boy has used is $24 \mathrm{~m}$ .

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Find the length of the kite string if the kite's height is $12 \mathrm{m}$ and the string makes a $30^{\circ}$ angle with the ground.

STEP 1

STEP 2

Draw a right-angled triangle to represent the situation. Label the height of the kite as the opposite side (O), the length of the string as the hypotenuse (H), and the angle between the string and the ground as $\theta = 30^{\circ}$ .

STEP 3

Use the trigonometric sine function, which relates the opposite side, the hypotenuse, and the angle in a right-angled triangle.
$\sin(\theta) = \frac{O}{H}$

STEP 4

Substitute the given values into the sine function.
$\sin(30^{\circ}) = \frac{12 \mathrm{~m}}{H}$

STEP 5

Calculate the value of $\sin(30^{\circ})$ .
$\sin(30^{\circ}) = \frac{1}{2}$

STEP 6

Substitute the value of $\sin(30^{\circ})$ into the equation.
$\frac{1}{2} = \frac{12 \mathrm{~m}}{H}$

STEP 7

Solve the equation for the hypotenuse (H) by multiplying both sides by H and then dividing by $\frac{1}{2}$ .
$H = \frac{12 \mathrm{~m}}{\frac{1}{2}}$

STEP 8

Calculate the length of the string (H).
$H = 12 \mathrm{~m} \times 2 = 24 \mathrm{~m}$
The length of the string that the boy has used is $24 \mathrm{~m}$ .

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Find the length of the kite string if the kite's height is 12m12 \mathrm{m}12m and the string makes a 30∘30^{\circ}30∘ angle with the ground.

STEP 1

STEP 2

Draw a right-angled triangle to represent the situation. Label the height of the kite as the opposite side (O), the length of the string as the hypotenuse (H), and the angle between the string and the ground as θ=30∘\theta = 30^{\circ}θ=30∘.

STEP 3

Use the trigonometric sine function, which relates the opposite side, the hypotenuse, and the angle in a right-angled triangle.sin⁡(θ)=OH\sin(\theta) = \frac{O}{H}sin(θ)=HO​

STEP 4

Substitute the given values into the sine function.sin⁡(30∘)=12 mH\sin(30^{\circ}) = \frac{12 \mathrm{~m}}{H}sin(30∘)=H12 m​

STEP 5

Calculate the value of sin⁡(30∘)\sin(30^{\circ})sin(30∘).sin⁡(30∘)=12\sin(30^{\circ}) = \frac{1}{2}sin(30∘)=21​

STEP 6

Substitute the value of sin⁡(30∘)\sin(30^{\circ})sin(30∘) into the equation.12=12 mH\frac{1}{2} = \frac{12 \mathrm{~m}}{H}21​=H12 m​

STEP 7

Solve the equation for the hypotenuse (H) by multiplying both sides by H and then dividing by 12\frac{1}{2}21​.H=12 m12H = \frac{12 \mathrm{~m}}{\frac{1}{2}}H=21​12 m​

STEP 8

Calculate the length of the string (H).H=12 m×2=24 mH = 12 \mathrm{~m} \times 2 = 24 \mathrm{~m}H=12 m×2=24 mThe length of the string that the boy has used is 24 m24 \mathrm{~m}24 m.

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Find the length of the kite string if the kite's height is $12 \mathrm{m}$ and the string makes a $30^{\circ}$ angle with the ground.

Draw a right-angled triangle to represent the situation. Label the height of the kite as the opposite side (O), the length of the string as the hypotenuse (H), and the angle between the string and the ground as $\theta = 30^{\circ}$ .

Use the trigonometric sine function, which relates the opposite side, the hypotenuse, and the angle in a right-angled triangle.
$\sin(\theta) = \frac{O}{H}$

Substitute the given values into the sine function.
$\sin(30^{\circ}) = \frac{12 \mathrm{~m}}{H}$

Calculate the value of $\sin(30^{\circ})$ .
$\sin(30^{\circ}) = \frac{1}{2}$

Substitute the value of $\sin(30^{\circ})$ into the equation.
$\frac{1}{2} = \frac{12 \mathrm{~m}}{H}$

Solve the equation for the hypotenuse (H) by multiplying both sides by H and then dividing by $\frac{1}{2}$ .
$H = \frac{12 \mathrm{~m}}{\frac{1}{2}}$

Calculate the length of the string (H).
$H = 12 \mathrm{~m} \times 2 = 24 \mathrm{~m}$
The length of the string that the boy has used is $24 \mathrm{~m}$ .