Question

One side of a right triangle is known to be 12 cm long and the opposite angle is measured as 30°, with a possible error of ±1°.(a) Use differentials to estimate the error in computing the length of the hypotenuse. (Round your answer to two decimal places.)______± cm(b) What is the percentage error? (Round your answer to the nearest integer.)± ______%

Answer 1

Answer:

a) ±0.73

b) ±3%

Step-by-step explanation:

One side of the right angle triangle is 12cm

The opposite angle(x) is 30 degree

Possible error = ±1degree

a) Let the hypotenus= h

h/12 = cosecx

h = 12cosecx

Differentiate h with respect to x

dh= -12cosecxcotcdx

The possible error in the angle measurement is dx

dx= ±1degree

Convert degree to radians

dx = ±1*π/180

dx = ± π/180

Put x and dx into dh

dh = -12cosec30cot30(±π/180)

= -12(2)(√3)(±π/180)

= -24√3(±π/180)

= ±24π√3/180

= ±0.73

b) To find the percentage error, recall that

dh = 12cosecx

dh = 12cosec30

dh = 12(2)

= 24

Percentage error =

(change in value/initial value) 100

Percentage error = (dh/h)100

= (±0.73/24)100

= ±0.030*100

= ±3%

Answer 2

Percentage error is simply the difference between the approximated and actual values, as a percentage of the actual value

• The error in the length of the hypotenuse is $\mathbf{\pm 0.73}$
• The percentage error is 3%

The relationship between the hypotenuse, and the opposite side of a triangle is:

$\mathbf{sin(\theta) = \frac{Opposite}{Hypotenuse}}$

So, we have:

$\mathbf{sin(30) = \frac{12}{h}}$

Take inverse of both sides

$\mathbf{csc(30) = \frac{h}{12}}$

Make h the subject

$\mathbf{h = 12csc(30) }$

Differentiate

$\mathbf{h' = -12csc(30)cot(30) x' }$

Express x' as radian

$\mathbf{h' = -12csc(30)cot(30) \times \frac{\pm \pi}{180} }$

$\mathbf{h' = -12\times 2 \times\sqrt 3 \times \frac{\pm \pi}{180} }$

$\mathbf{h' = -\sqrt 3 \times \frac{\pm \pi}{7.5} }$

$\mathbf{h' = \pm 0.73}$

Hence, the error in the length of the hypotenuse is $\mathbf{\pm 0.73}$

(b) The percentage error

In (a), we have:

$\mathbf{h = 12csc(30) }$

$\mathbf{h = 12 \times 2}$

$\mathbf{h = 24}$

So, the percentage error is:

$\mathbf{\% Error = \frac{h'}{h} \times 100\%}$

This gives

$\mathbf{\% Error = \frac{\pm 0.73}{24} \times 100\%}$

$\mathbf{\% Error = \frac{\pm 73}{24} \%}$

$\mathbf{\% Error = \pm 3 \%}$

Hence, the percentage error is 3%

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