Byron wanted to try out different watercraft. He went 124 miles downstream in a motorboat and 45 miles downstream on a jet ski. His speed on the jet ski was 10 mph faster than in the motorboat. Byron spent a total of 4 hours on the water. What was his rate of speed, in miles per hour, in the motorboat?

the speed of the motorboat was 90.313 mph

Explanation:

since the distance d covered for each watercraft is

d= v*t , where v= velocity and t = time spent in that watercraft

the difference in speed will be

v₂-v₁ = d₂/t₂ - d₁/t₁ = 10 mph

with the constraint

t₂ + t₁ = 4 hours → t₂ = 4- t₁

where 2 represents motorboat and 1 jet ski

then

d₂/t₂ - d₁/t₁ = 10

d₂/(4- t₁) - d₁/t₁= 10

d₂*t₁ - d₁*(4- t₁) = 10 * t₁ * (4- t₁)

d₂*t₁ - 4*d₁ + d₁*t₁ = 40*t₁ - 10*t₁²

10*t₁² + (d₂+d₁-40) *t₁ - 4*d₁ = 0

replacing values of distances

10*t₁² + (d₂+d₁-40) *t₁ - 4*d₁ = 0

10*t₁² + (45+124- 40) *t₁ - 4*124 = 0

10*t₁² +129 *t₁ - 496 = 0

then

t₁ = [-129 + √(129²+4*10*496)]/(2*10) = 1.373 hours

therefore

v₁ = d₁/t₁ = 124 miles/1.373 hours = 90.313 mph

thus the speed of the motorboat was 90.313 mph

change tension

see the picture for the vector

the magnitude of the vector by pythagorean theorem

r = √( (-10)^2 + (3)^2 ) = √109

magnitude = √109 units

the direction angle between vector and x-axis is

tan^(-1)(3/10) = 16.699

so the direction is approximately 16.7 degrees north of west