Ben's boat will take 1½ hours longer to go 12 miles up a river than to return. what is the rate of his boat if the rate of the current is 2 miles an hour?

Answer

= 4.8 miles per hours.

Explanation

Let the time taken to run down the river to be t hours.

The time the bout takes to run up the river will be (t+1.5) hours.

Let the rate of the boat to be x miles per hour.

So the speed of the boat up the river will be (x-2) miles per hour due to the drag the the current.

The speed of the boat down the river will be (x+2).

Distance = 12 miles

With this information we can form two equations and solve them simultaneously.

distance = speed × time

12 = (t+3/2)(x-2)

12 = tx-2t+1.5x-3

tx - 2t + 1.5x =15(i)

12 = t(x+2)

t = 12/(x+2)(ii)

Substitute t in equation (i) with 12/(x+2)

tx - 2t + 1.5x =15

12x/(x+2) - 2x/(x+2) +1.5x = 15

12x - 2x +1.5x(x+2) = 15(x+2)

10x + 1.5x²+3x=15x+30

1.5x²-2x-30 = 0

3x² - 2x - 60 = 0

Using quadratic formula;

x = (--2 (+ or -)√(2²+4×3×60))÷(2×3)

x = 4.8 or

x= -4.15.

Since speed cannot be negative, x = 4.8.

Rate of the boat = 4.8 miles per hours.

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for