Chapter 1, Problem 16Q

(a)

To determine

The distance at which a person will have to hold a European 2-euro coin to subtend an angle of $1°$.

Answer to Problem 16Q

Solution:

1.49 m

Explanation of Solution

Given data:

A European 2-euro coin has a diameter of 2.6 cm. The angle subtended by the coin is $1°$.

Formula used:

The small-angle formula is written as,

$D=\frac{\alpha d}{206,265}$

Here, α is the angle subtended by the object (in arcseconds), d is the distance between the observer and the object and D is the linear size of the object.

Explanation:

The distance at which a person will hold a 2-euro coin to subtend an angle of $1°$ can be calculated by using the small-angle formula, that is,

$D=\frac{\alpha d}{206,265}$

Rearrange for d,

$d=\frac{206265D}{\alpha }$

Since $1°$ is equal to $\text{3600}{″}^{}$, substitute 2.6 cm for D and $\text{3600}{″}^{}$ for α,

$\begin{array}{c}d=\frac{206265\left(2.6\text{ cm}\right)}{3600^{″}}\\ =\frac{536289}{3600\text{'}\text{'}}\\ =148.97\text{ cm}\left(\frac{1\text{ m}}{100\text{ cm}}\right)\\ =1.49\text{ m}\end{array}$

Conclusion:

Hence, the coin should be held at 1.49 m if the angle to be subtended is $1°$.

(b)

To determine

The distance at which a person will have to hold a European 2-euro coin to subtend an angle of $1\text{ arcmin}$.

Answer to Problem 16Q

Solution:

89.3815 m

Explanation of Solution

Given data:

A European 2-euro coin has a diameter of 2.6 cm. The angle subtended by the coin is $\text{1}{\prime }^{}\text{(arcminute)}$.

Formula used:

The small-angle formula is written as,

$D=\frac{\alpha d}{206,265}$

Here, α is the angle subtended by the object (in arcseconds), d is the distance between the observer and the object and D is the linear size of the object.

Explanation:

The distance at which a person will hold a 2-euro coin to subtend an angle of $1^{\prime }$ can be calculated by using the small-angle formula, that is,

$D=\frac{\alpha d}{206,265}$

Rearrange for d,

$d=\frac{206265D}{\alpha }$

Since $1^{\prime }$ is equal to $\text{60}{″}^{}$, substitute 2.6 cm for D and $\text{60}{″}^{}$ for α,

$\begin{array}{c}d=\frac{206265\left(2.6\text{ cm}\right)}{60^{″}}\\ =\frac{536289}{60^{″}}\\ =8938.15\text{ cm}\left(\frac{1\text{ m}}{100\text{ cm}}\right)\\ =89.4\text{ m}\end{array}$

Conclusion:

Hence, the coin should be held at 89.4 m in order to subtend an angle of $1^{\prime }$.

(c)

To determine

The distance at which a person will have to hold a European 2-euro coin to subtend an angle of $1\text{ arcsec}$.

Answer to Problem 16Q

Solution:

5362.9 m

Explanation of Solution

Given data:

A European 2-euro coin that has a diameter of 2.6 cm. The angle subtended by the coin is $1^{″}$.

Formula used:

The small-angle formula is written as,

$D=\frac{\alpha d}{206,265}$

Here, α is the angle subtended by the object (in arcseconds), d is the distance between the observer and the object and D is the linear size of the object.

Explanation:

The distance at which a person will hold a 2-euro coin to subtend an angle of $1^{″}$ can be calculated by using the small-angle formula, that is,

$D=\frac{\alpha d}{206,265}$

Rearrange for d,

$d=\frac{206265D}{\alpha }$

Substitute 2.6 cm for D and $1^{″}$ for α,

$\begin{array}{c}d=\frac{206265\left(2.6\text{ cm}\right)}{1^{″}}\\ =\frac{536289}{1\text{'}\text{'}}\\ =536289\text{cm}\left(\frac{1\text{ m}}{100\text{ cm}}\right)\\ =5362.9\text{ m}\end{array}$

Conclusion:

Hence, the coin should be held at 5362.9 m in order to subtend an angle of $1^{″}$.