Question

The mass of a newborn baby's brain has been found to increase by about \(1.6 \mathrm{mg}\) per minute. (a) How much does the brain's mass increase in 1 day? (b) How long does it take for the brain's mass to increase by \(0.0075 \mathrm{~kg}\) ?

Short answer

(a) 2.304 g in 1 day; (b) 3.255 days for a 0.0075 kg increase.

Step-by-step solution

Convert the mass increase rate to per day

The brain's mass increases by \(1.6 \mathrm{mg}\) per minute. First, calculate the number of minutes in one day: \(24 \times 60 = 1440\) minutes. Then, multiply the rate of increase by the number of minutes in a day to find the total mass increase in one day: \(1.6 \mathrm{mg/min} \times 1440 \mathrm{min} = 2304 \mathrm{mg/day}\).

Convert the mass increase from milligrams to grams

Since \(1 \mathrm{mg} = 0.001 \mathrm{g}\), convert the daily increase from milligrams to grams: \(2304 \mathrm{mg} \times 0.001 = 2.304 \mathrm{g}\).

Convert the target mass increase to milligrams

The target mass increase is \(0.0075 \mathrm{kg}\). Convert kilograms to milligrams: \(0.0075 \mathrm{kg} = 7500 \mathrm{mg}\).

Calculate the time to reach the target increase

To find out how long it takes for the brain's mass to increase by \(7500 \mathrm{mg}\), divide the target increase by the rate of increase: \(\frac{7500 \mathrm{mg}}{1.6 \mathrm{mg/min}} = 4687.5 \mathrm{min}\).

Convert minutes to hours and days

Convert the time from minutes to hours by dividing by \(60\): \(\frac{4687.5}{60} \approx 78.125 \mathrm{hours}\). Then convert hours to days by dividing by \(24\): \(\frac{78.125}{24} \approx 3.255 \mathrm{days}\).

Key concepts

Mass Conversion

When dealing with measurements, it's all about ensuring that the units are compatible, so calculations make sense. In the brain growth exercise, we need to switch between milligrams, grams, and kilograms. These units form part of the metric system, where the conversions are straightforward once you know the basic principles.

To convert from one unit to another in this system, you use powers of ten. For instance, converting milligrams (mg) to grams (g), you need to know that 1 gram equals 1000 milligrams. Thus, to switch mg to g, you multiply by 0.001 since 1 mg is 0.001 grams.

Similarly, going from kilograms (kg) to milligrams involves multiplying by 1,000,000 because 1 kg is 1,000,000 mg. The conversion steps ensure you're using the right scale, which is key in accurately determining how much a newborn baby's brain grows in mass.

Units of Measurement

Units of measurement are critical in any scientific calculation as they help ensure precision and clarity. In the original exercise, the brain's mass increase was gauged in milligrams per minute.

Switching between different units like milligrams and kilograms requires an understanding of these units' significance and their respective scales within the metric system.

Each unit change should make logical sense in terms of size and relevance. For these calculations, milligrams are smaller than grams, and kilograms are larger, suitable for different precision levels in measurements.

• Milligrams (mg) - Suitable for measuring small masses, perfect for slight brain growth increments.
• Grams (g) - A middle ground, perfect for daily increases.
• Kilograms (kg) - Used for larger calculations and more significant weight differences.

Understanding the relationship between these scales helps us convert and calculate changes accurately and efficiently.

Time Calculation

Time calculation is essential for understanding change over periods. In observing brain growth, the rate of change per minute is considered. To extend this to a longer period, such as a day, you need calculations that can translate minutes into hours or days.

We know there are 60 minutes in an hour and 24 hours in a day. To find how many minutes are in a day, you multiply: 24 hours × 60 minutes = 1440 minutes. This gives you the total minutes available for growth calculation per day.

• Minutes – Used for highly precise short intervals.
• Hours – Bridging smaller and larger time frames, useful for moderate computations.
• Days – Great for overarching trends or increases over extended periods.

Once you calculate these total periods, you can apply them to the mass changes to determine growth accurately over larger spans. This ensures the overall time calculation remains consistent with all units use when determining the brain's growth in mass.