Question

1. an average human heart beats 60 times per minute. if the average person lives to the age 85, how many times does the average heart beat in a lifetime?
2. blood sugar levels are measured in milligrams of glucose per decilitre of blood volume. if a person’s blood sugar level measured 128 mg/dl, what is this in grams per litre?
3. the following lengths are given in metres. use metric prefixes to rewrite them so the numerical value is bigger than one but less than 1000. for example, $7.9 \times 10^{-2}$ m could be written either as 7.9 cm or 79 mm. (a) $7.59 \times 10^{7}$ m;

(b) 0.0074 m; (c) $8.8 \times 10^{-11}$m;
(d) $1.63 \times 10^{13}$ m.

1. the volume of earth is on the order of $10^{21}$ m³. (a) what is this in cubic kilometres (km³)?

(b) what is it in cubic centimetres (cm³)?

1. the following masses are given in kilograms. use metric prefixes on the gram to rewrite them so the numerical value is bigger than one but less than 1000. for example, $7 \times 10^{-5}$ kg could be written as 70 cg or 700 mg. (a) $3.8 \times 10^{-5}$kg; (b) $2.3 \times 10^{17}$kg; (c) $2.4 \times 10^{-11}$ kg

(d) $8 \times 10^{15}$ kg; (e) $4.2 \times 10^{-3}$ kg.

1. a car is traveling at a speed of 33 m/s. (a) what is its speed in kilometres per hour?

(b) is it exceeding the 80 km/h speed limit?

Explanation:

Step1: Solve Q1 (Diamond mass)

Given 1 carat = 0.20 grams, total carats = 3106.
$\text{Mass} = 3106 \times 0.20 = 621.2$ grams

Step2: Solve Q2 (Lifetime heartbeats)

Convert 83 years to minutes: $83 \times 365 \times 24 \times 60 = 43624800$ minutes.
$\text{Total beats} = 43624800 \times 60 = 2.617488 \times 10^9$

Step3: Solve Q3 (Blood sugar conversion)

1 mg = $10^{-3}$ g, 1 dL = $10^{-1}$ L.
$\text{Conversion: } 128 \times \frac{10^{-3}\text{ g}}{10^{-1}\text{ L}} = 1.28$ g/L

Step4: Solve Q4 (Metric prefix rewrite)

(a) $7.59 \times 10^7$ m = $75.9 \times 10^6$ m = 75.9 Mm
(b) $0.0074$ m = $7.4 \times 10^{-3}$ m = 7.4 mm
(c) $8.8 \times 10^{-11}$ m = $88 \times 10^{-12}$ m = 88 pm
(d) $1.63 \times 10^{13}$ m = $16.3 \times 10^{12}$ m = 16.3 Tm

Step5: Solve Q5 (Earth volume conversion)

(a) $1$ km = $10^3$ m, so $1$ km³ = $(10^3)^3$ m³ = $10^9$ m³.
$\text{Volume} = \frac{10^{21}}{10^9} = 10^{12}$ km³
(b) $1$ m = $10^2$ cm, so $1$ m³ = $(10^2)^3$ cm³ = $10^6$ cm³.
$\text{Volume} = 10^{21} \times 10^6 = 10^{27}$ cm³

Step6: Solve Q6 (Mass metric prefix rewrite)

(a) $3.8 \times 10^{-5}$ kg = $3.8 \times 10^{-2}$ g = 3.8 cg
(b) $2.3 \times 10^{17}$ kg = $2.3 \times 10^{20}$ g = 230 Eg
(c) $2.4 \times 10^{-11}$ kg = $2.4 \times 10^{-8}$ g = 24 ng
(d) $8 \times 10^{15}$ kg = $8 \times 10^{18}$ g = 8 Eg
(e) $4.2 \times 10^{-3}$ kg = 4.2 g

Step7: Solve Q7 (Car speed conversion)

(a) 1 m/s = 3.6 km/h.
$\text{Speed} = 33 \times 3.6 = 118.8$ km/h
(b) Compare to 80 km/h: $118.8 > 80$

Answer:

1. 621.2 grams
2. $2.62 \times 10^9$ beats (rounded)
3. 1.28 g/L
4. (a) 75.9 Mm; (b) 7.4 mm; (c) 88 pm; (d) 16.3 Tm
5. (a) $10^{12}$ km³; (b) $10^{27}$ cm³
6. (a) 3.8 cg; (b) 230 Eg; (c) 24 ng; (d) 8 Eg; (e) 4.2 g
7. (a) 118.8 km/h; (b) Yes, it is exceeding the speed limit