#1. After 16 nights the penny will double 16 times=2^16 or 65,536. That's a lot of pennies! 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024, 2,048, 4,096, 8,192, 16,384, 32,768, 65,536. The child couldn't sleep with all those under its pillow, and they would be worth $655.36 or £655.36, depending on whether it's a cent or a UK penny.
#2. An acre is 43,560 sq ft. The third side of the triangle is undefined so I'm going to assume it's a right-angled triangle. The property line is just the boundary line. I'm going to assume that the sides that include the right angle are 170 and 900, so the area in sq ft is 1/2*base*height=0.5*170*900=76500sq ft=76500/43560 acres=1.756 acres.
#3. Circumference=(pi)*d=12(pi)=37.7cm and area is (pi)*(d/2)^2=36(pi)=113.1 sq cm.
#4. Sound spreads out in spherical waves. The same amount of sound has to be spread around the surface area of a sphere no matter how big the sphere is, and the intensity of the sound is inversely proportional to the surface area of the sphere. At 7 metres the surface area will be 49 times what it is at 1 metre, so the sound is 49 times less intense at 7 metres. Therefore the sound will be 49 times as intense at 1 metre as it is at 7 metres.
#5. At 35 mph the distance travelled is speed*time. So after 4.3hrs=150.5 miles and after 7.7hrs=269.5 miles.
#6. Half-life is when encroachment has reached 50% or a half. What we have to solve is 0.95^x=0.5 where x is the number of years. We need logs: xlog0.95=log0.5. Let's use logs to base 2 x=-1/-0.074=13.5 years approx. In 10 years encroachment would reduce the forest to 59.9% (0.95^10 as a percentage).
#7. After 5 seconds, half the original amount remains=40g. After a further 5 secs half as much remains=20g; after 15 seconds, 10g; after 20 secs, 5g; and after 25 secs, 2.5g.