5. An astronomical unit (au) is a unit of length defined as 149597870700 metres, which is the approximate average distance between the Earth and the Sun. (i) Express this astronomical unit in metres, correct to 4 significant figures. Give your answer in standard form. (ii) The speed of light is $$ 3 imes 10^{8} \mathrm{~m} / \mathrm{s} $$. How long

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5. An astronomical unit (au) is a unit of length defined as 149597870700 metres, which is the approximate average distance between the Earth and the Sun. (i) Express this astronomical unit in metres, correct to 4 significant figures. Give your answer in standard form. (ii) The speed of light is $$ 3 	imes 10^{8} \mathrm{~m} / \mathrm{s} $$. How long

UpStudy AI · Accepted Answer

Let's solve the problem step by step.

### Part (i)
We need to express the astronomical unit (au) in meters, correct to 4 significant figures, and in standard form.

The value of 1 astronomical unit is given as:
$$
149597870700 \text{ metres}
$$

To express this in standard form and to 4 significant figures, we first identify the significant figures in the number.

1. The first four significant figures of $$ 149597870700 $$ are $$ 1495 $$.
2. The number can be expressed in standard form as $$ 1.49597870700 \times 10^{11} $$.

Now, rounding to 4 significant figures gives us:
$$
1.496 \times 10^{11} \text{ metres}
$$

### Part (ii)
Next, we need to calculate how long it takes for light to travel 1 astronomical unit (au) at the speed of light, which is $$ 3 \times 10^{8} \text{ m/s} $$.

To find the time $$ t $$, we can use the formula:
$$
t = \frac{d}{v}
$$
where:
- $$ d $$ is the distance (1 au in meters),
- $$ v $$ is the speed of light.

Substituting the values:
$$
t = \frac{149597870700 \text{ m}}{3 \times 10^{8} \text{ m/s}}
$$

Now, let's calculate this value.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{149597870700}{3\times 10^{8}}$$
- step1: Rewrite the expression:
  $$\frac{1.495978\times 10^{11}}{3\times 10^{8}}$$
- step2: Rewrite the expression:
  $$\frac{1495978707\times 10^{2}}{3\times 10^{8}}$$
- step3: Reduce the fraction:
  $$\frac{498659569}{10^{6}}$$
The time it takes for light to travel 1 astronomical unit (au) is approximately:

$$
t = 498.659569 \text{ seconds}
$$

In standard form, this can be expressed as:
$$
4.9866 \times 10^{2} \text{ seconds}
$$

### Summary of Answers:
(i) The astronomical unit in meters, correct to 4 significant figures, is:
$$
1.496 \times 10^{11} \text{ metres}
$$

(ii) The time taken for light to travel 1 astronomical unit is:
$$
4.9866 \times 10^{2} \text{ seconds}
$$
The astronomical unit is $$1.496 \times 10^{11}$$ metres. Light takes approximately $$4.9866 \times 10^{2}$$ seconds to travel 1 astronomical unit.

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