Question

the drag Force F on a boat varies jointly with the wet surface area a of the boat and the square of the speed s of the boat. a boat with a wet surface area of 50 ft^2 traveling at 7 miles per hour experiences a drag force of 98 N. find the wet surface area of a boat traveling 7.5 miles per hour and experiencing a drag force of 135 N.__ft^2

Answer 1

Let F be the drag force, A the area and s the speed. We are told that F varies jointly with respect to s and A. This means that we have an equation of the form

$F=k\cdot A\cdot s$

where k is a constant. We are told that when A=50 and s=7, then F=98. So we have the equation

$98=k\cdot50\cdot7=k\cdot350$

If we divide both sides by 350, we get

$k=\frac{98}{350}=\frac{49\cdot2}{175\cdot2}=\frac{49}{175}=\frac{7\cdot7}{25\cdot7}=\frac{7}{25}$

So our equation becomes

$F=\frac{7}{25}\cdot A\cdot s$

Now, we want to calculate A, when s=7.5 and F=135. so we have the equation

$135=\frac{7}{25}\cdot A\cdot7.5$

Note that 7.5=15/2. So we have

$135=\frac{7}{25}\cdot A\cdot\frac{15}{2}=\frac{7\cdot5\cdot3}{5\cdot5\cdot2}\cdot A=\frac{7\cdot3}{5\cdot2}\cdot A=\frac{21}{10}\cdot A$

Now, we multiply both sides by 10, so we get

$21A=135\cdot10=1350$

Finally, by dividng both sides by 21, we get

$A=\frac{1350}{21}=64.2857$

so the wet surface area is approximately 64.29 ft^2