#### To determine

**To estimate:** The error in computing the length of the hypotenuse by using differential.

#### Answer

The maximum error to computing the length of the hypotenuse is dx≈±1.21 cm.

#### Explanation

**Given:**

One side of the triangle is 20 cm and opposite angle is 30° with error is ±1°.

**Calculation:**

Obtain the hypotenuse of the triangle.

Form Figure 1, it is observed that

sinθ=20xx=20sinθx=20cscθ

The differential is dx=f′(θ)dθ.

The derivative of the function f(θ)=20cscθ is computed as follows,

f′(θ)=ddθ(cscθ)=20ddθ(cscθ)=20(−cotθcscθ)=−20cotθcscθ

Substitute the f′(θ)=−20cotθcscθ in dx=f′(θ)dθ,

dx=(−20cotθcscθ)dθ

Substitute the value θ=30° and dθ=±1°,

dx=(−20cot30°csc30°)(±1°)=−20(3)(2)(±π180) (Q1°=π180)=±23π9≈±1.21 cm

Therefore, the maximum error to computing the length of the hypotenuse is dx≈±1.21 cm.

#### To determine

**To find:** The percentage error.

#### Answer

The percentage error is ±3%.

#### Explanation

**Calculation:**

Obtain the percentage error.

The relative error Δxx is computed as follows,

Δxx≈dxx

Substitute the value dx≈±1.21 cm (from part (a)) and x=20csc30°,

Δxx≈±1.2120csc30°=±1.2120(2)=±0.03

Thus, the relative error is ±0.03.

Since the percentage error is product of relative error and 100%, ±0.03×100%=3%.

Therefore, the percentage error is ±3%.