Speed, density and pressure

Distance $=$ speed $\times$ time

$37 \times 3 = 111$ miles

45 minutes $= \frac{45}{60} = 0.75$ hours, so the time is $1.75$ hours.

Distance = speed × time $= 56 \times 1.75 = 98$ km

(a) Distance = speed × time $= 54 \times 1.5 = 81$ miles

(b) Real length $= 6 \times 25\,000 = 150\,000$ cm

$150\,000 \div 100 = 1500$ m, and $1500 \div 1000 = 1.5$ km

(a) Time $= 50 \div 40 = 1.25$ hours

$0.25 \times 60 = 15$, so 1 hour 15 minutes.

07 30 + 1 h 15 min = 08 45

(b) Going faster over the same distance takes less time, so he arrives earlier.

(a) 9 am to 10 30 am is 1.5 hours.

Distance $= 4 \times 1.5 = 6$ miles

(b) Leaves library: 10 30 + 50 min = 11 20 am

$1\frac{1}{4}$ hours is 1 hour 15 minutes: 11 20 + 1 h 15 min = 12 35 pm

Minutes parked: $8.40 \div 0.024 = 350$ minutes

$350$ minutes $= 5$ hours $50$ minutes

$10{:}45 + 5\text{ h }50\text{ min} = 16{:}35$

(a) Speed $= \frac{\text{distance}}{\text{time}} = 186 \div 3 = 62$ mph

(b) Distance $=$ speed $\times$ time $= 58 \times 4 = 232$ miles

(a) Speed $= 36 \div 3 = 12$ km/h

(b) Tuesday time $= 36 \div 16 = 2.25$ hours $= 2$ h $15$ min

Total $= 3$ h $+ 2$ h $15$ min $= 5$ h $15$ min

$5$ h $15$ min $< 5$ h $20$ min, so Lizzie is correct.

(a) Round: $4.96 \approx 5$ miles, $30.4 \approx 30$ mph

Time $= 5 \div 30 = \frac{1}{6}$ hour $= 10$ minutes

(b) The distance was rounded up and the speed rounded down, so the estimate is too big: an overestimate.

UK hourly pay: $473.28 \div 32 = £14.79$

Australian hourly pay in pounds: $26.40 \div 1.796 = £14.699...$

$£14.70 < £14.79$, so no, Blake is not correct.

Car time $= 143 \div 55 = 2.6$ hours $= 2$ hours 36 minutes ($0.6 \times 60 = 36$).

Total $= 2$ h 36 min $+ 5$ h 20 min $= 7$ hours 56 minutes

Seconds: $67\,205\,600 \div 11.9 = 5\,647\,529.4...$

Seconds in a day: $24 \times 60 \times 60 = 86\,400$

$5\,647\,529.4 \div 86\,400 = 65.36...$, so $65$ days

Mass = density × volume

$9 \times 72 = 648$ g

Cycling: $30 \div 24 = 1.25$ h

Running: $12 \div 8 = 1.5$ h

Total: $2.75$ h $= 2$ hours $45$ minutes

Time: 13 30 to 14 48 is 1 hour 18 minutes $= 78$ minutes.

In hours: $78 \div 60 = 1.3$

Speed $= 65 \div 1.3 = 50$ mph

(a) 1 min 54 s $= 114$ s

Time $= \frac{700}{475} \times 114 = 168$ s

$168$ s $= 2$ minutes $48$ seconds

(b) If she speeds up over a longer race, she will finish more quickly, so the real time is less than 2 min 48 s.

$p = \frac{F}{A}$, so $A = \frac{F}{p}$.

$A = \frac{140}{35} = 4$ m²

Area $= \text{force} \div \text{pressure} = 720 \div 80 = 9\text{ m}^2$

Length $= 9 \div 2 = 4.5$ m

Volume $= \pi \times 3^2 \times 10 = 90\pi = 282.74...\text{ cm}^3$

Density $= 2250 \div 282.74... = 7.957...$

$= 7.96\text{ g/cm}^3$ (3 s.f.)

Density = mass ÷ volume

$27 \div 10 = 2.7$ g/cm³

Volume = base area × height, so base area $= 1200 \div 40 = 30$ cm²

Pressure $= \frac{90}{30} = 3$ newtons/cm²

Area of face on the table $= \text{force} \div \text{pressure} = 504 \div 3.5 = 144$ cm²

A cube has 6 equal faces: $6 \times 144 = 864$ cm²

$864 < 900$, so the surface area is less than 900 cm².

Area $= 4 \times 2 = 8\text{ m}^2$

Pressure $= \frac{10\,000}{8} = 1250$ N/m$^2$

Volumes: $3^3 = 27$ cm³ and $4^3 = 64$ cm³

Densities: $81 \div 27 = 3$ g/cm³ and $128 \div 64 = 2$ g/cm³

Ratio: $3 : 2$

Area $= \text{force} \div \text{pressure} = 180 \div 187.5 = 0.96\text{ m}^2$

Area $= 1.2 \times x$, so $x = 0.96 \div 1.2 = 0.8$

Stage 1: $6 \times \frac{15}{60} = 1.5$ miles

Stage 2: $9 \times \frac{40}{60} = 6$ miles

Total = 7.5 miles

Amy: $7.5 \div 0.75 = 10$ mph

In one hour: $30 \times 3600 = 108\,000$ metres

$108\,000 \div 1000 = 108$ km/h

Mass of A $= 1.8 \times 80 = 144$ g. Mass of B $= 1.2 \times 40 = 48$ g.

Total mass $= 192$ g, total volume $= 120$ cm³.

Density $= 192 \div 120 = 1.6$ g/cm³.

Company A: 1 min 40 s $= \frac{5}{3}$ min, so the rate is $8 \div \frac{5}{3} = 4.8$ litres per minute.

Time A $= 2400 \div 4.8 = 500$ min.

Company B: $2.2 \times 4.54 = 9.988$ litres per minute.

Time B $= 2400 \div 9.988 = 240.29$ min.

Difference $= 500 - 240.29 = 259.7$, about 260 minutes.

Base area $= \frac{18}{3} = 6\text{ m}^2$.

Force $=$ pressure $\times$ area $= 75 \times 6 = 450$ N.

Triangle area: $\frac{1}{2} \times 4 \times 6 = 12$ cm²

Volume: $12 \times 10 = 120$ cm³

Mass: $120 \times 0.8 = 96$ g

(a) 18 minutes $= \frac{18}{60} = 0.3$ hours

Distance $= 72 \times 0.3 = 21.6$ km

(b) $72$ km/h $= 72\,000$ m in $3600$ s

$72\,000 \div 3600 = 20$ m/s. The speeds are the same, so David is not correct.

Rearrange $p = \frac{F}{A}$ to $A = \frac{F}{p}$.

$A = \frac{140}{35} = 4$

The area is $4$ m².

First part: $50 \times 3 = 150$ miles.

Second part: $150 \div 30 = 5$ hours.

Total: $300$ miles in $8$ hours.

Average speed $= 300 \div 8 = 37.5$ mph, so Sean is wrong.

Density of B is 1.1 g/cm³, so density of A $= \frac{19}{22} \times 1.1 = 0.95$ g/cm³.

Mass of C $= 5 \times 0.95 + 15 \times 1.1 = 4.75 + 16.5 = 21.25$ g.

Density of C $= 21.25 \div 20 = 1.0625$ g/cm³.

Volume of tin $= \frac{\text{mass}}{\text{density}} = 600 \div 0.6 = 1000$ cm³

Volume of salt $= 1000 - 700 = 300$ cm³

Density of salt $= 600 \div 300 = 2$ g/cm³

Cylinder volume $= \pi \times 3^2 \times 8 = 226.19…$ cm³

Mass $= 226.19… \times 7.86 = 1777.89…$ g

Cube volume $= 1777.89… \div 8.5 = 209.16…$ cm³

$h = \sqrt[3]{209.16…} = 5.936… \approx 5.9$

Volume of A $= \frac{1400}{70} = 20\text{ m}^3$

Mass of B $= 280 \times 30 = 8400$ kg

Density of C $= \frac{1400 + 8400}{20 + 30} = \frac{9800}{50} = 196\text{ kg/m}^3$

Work per cm³ scaled by litres (the conversion factor cancels):

Ethanol: $1.09 \times 60 = 65.4$

Propylene: $0.97 \times 128 = 124.16$

Density $= \frac{65.4 + 124.16}{188} = \frac{189.56}{188} = 1.008... \approx 1.01$ g/cm³

Volume $= \pi \times 3^2 \times 25 = 225\pi = 706.86$ cm³

A: $\frac{2}{15} \times 225\pi = 30\pi$ cm³, mass $= 30\pi \times 1.21 = 114.04$ g

B: $\frac{13}{15} \times 225\pi = 195\pi$ cm³, mass $= 195\pi \times 1.02 = 624.86$ g

Total $= 738.90... \approx 739$ g

Top volume: $92.8 \div 2.9 = 32\text{ cm}^3$

Total mass: $92.8 + 972.8 = 1065.6$ g

Total volume: $1065.6 \div 4.7 \approx 226.72\text{ cm}^3$

Percentage: $32 \div 226.72 \times 100 \approx 14.1\%$