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Maria was travelling in her boat . . ., two equations – three unknowns

by VIJAY
(Chandigarh)











































My Question is written below:

"MARIA WAS TRAVELLING IN HER BOAT WHEN THE WIND BLEW HER HAT OFF AND HAT STARTED FLOATING AT DOWNSTREAM. THE BOAT CONTINUED TO TRAVEL UPSTREAM FOR 12 MORE MINUTES BEFORE MARIA REALIZED THAT HER HAT HAD FALLEN OFF AND TURNED BACK DOWNSTREAM. SHE CAUGHT UP WITH THAT AS SOON AS IT REACHED THE STARTING POINT. FIND THE SPEED OF RIVER IF MARIA'S HAT FLEW OFF EXACTLY 3 KM FROM WHERE SHE STARTED"

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Oct 19, 2011
Linear Equation in Two Variables
by: Staff

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Part II

Note that there are three unknowns, but only two equations. Therefore, the problem may have multiple solutions (it doesn’t, but you don’t know that at this point).

Solve for t in equation 2:

(t + 0.2) * R = 3

Rt + 0.2R = 3

Rt + 0.2R - 0.2R = 3 - 0.2R

Rt + 0 = 3 - 0.2R

Rt = 3 - 0.2R

t = (3 - 0.2R) / R


substitute the equation for t into the first equation:

t * (B + R) = 0.2 * (B - R) + 3

t * (B + R) = 0.2 * (B - R) + 3

[(3 - 0.2R) / R] * (B + R) = 0.2 * (B - R) + 3



Solve for R.


[(3 - 0.2R) / R] * (B + R) = 0.2 * (B - R) + 3


[(3 - 0.2R) * (B + R) / R] = 0.2 * B - 0.2 * R + 3

[(3 - 0.2R) * (B + R) / R] * R = (0.2 * B - 0.2 * R + 3) * R

(3 - 0.2R) * (B + R) * (R / R) = (0.2 * B - 0.2 * R + 3) * R

(3 - 0.2R) * (B + R) * (1) = (0.2 * B - 0.2 * R + 3) * R

(3 - 0.2R) * (B + R) = (0.2 * B - 0.2 * R + 3) * R

3B + 3R - 0.2R*B - 0.2R*R = (0.2 * B - 0.2 * R + 3) * R

3B + 3R - 0.2BR - 0.2R² = (0.2 * B - 0.2 * R + 3) * R

3B + 3R - 0.2BR - 0.2R² = 0.2 * B * R - 0.2 * R * R + 3 * R

3B + 3R - 0.2BR - 0.2R² + 0.2R² = 0.2BR - 0.2R² + 0.2R² + 3R

3B + 3R - 0.2BR + 0 = 0.2BR + 0 + 3R

3B + 3R - 0.2BR = 0.2BR + 3R

3B + 3R - 3R - 0.2BR = 0.2BR + 3R - 3R

3B + 0 - 0.2BR = 0.2BR + 0

3B - 0.2BR = 0.2BR

0.2BR = 3B - 0.2BR

(0.2BR) / B = (3B - 0.2BR) / B

0.2R = 3 - 0.2R

0.2R + 0.2R = 3 - 0.2R + 0.2R

0.2R + 0.2R = 3

0.4R = 3

(0.4R) / 0.4 = 3 / 0.4

R = 3 / 0.4

R = 7.5 km per hour

The final answer is: the speed of the river is 7.5 km per hour



What about the speed of the boat?

Solve for t

(t + 0.2) * R = 3

Rt + 0.2R = 3

Rt + 0.2R - 0.2R = 3 - 0.2R

Rt + 0 = 3 - 0.2R

Rt = 3 - 0.2R

(Rt) / R = (3 - 0.2R) / R

t = (3 - 0.2R) / R

R = 7.5

t = (3 - 0.2 * 7.5) / 7.5

t = 0.2 hours


t * (B + R) = 0.2 * (B - R) + 3

0.2 * (B + 7.5) = 0.2 * (B - 7.5) + 3

0.2 * B + 0.2 * 7.5 = 0.2 * B - 0.2 * 7.5 + 3

0.2B + 1.5 = 0.2 * B - 1.5+ 3

0.2B + 1.5 = 0.2B + 1.5


The speed of the boat can be any real number:

B ∈ ℝ


In summary:

R = 7.5 km per hour

B ∈ ℝ


Thanks for writing.

Staff
www.solving-math-problems.com



Oct 19, 2011
Linear Equation in Two Variables
by: Staff


Part I

Question:

by VIJAY
(Chandigarh)

My Question is written below:

"MARIA WAS TRAVELLING IN HER BOAT WHEN THE WIND BLEW HER HAT OFF AND HAT STARTED FLOATING AT DOWNSTREAM. THE BOAT CONTINUED TO TRAVEL UPSTREAM FOR 12 MORE MINUTES BEFORE MARIA REALIZED THAT HER HAT HAD FALLEN OFF AND TURNED BACK DOWNSTREAM. SHE CAUGHT UP WITH THAT AS SOON AS IT REACHED THE STARTING POINT. FIND THE SPEED OF RIVER IF MARIA'S HAT FLEW OFF EXACTLY 3 KM FROM WHERE SHE STARTED"


Answer:

B = speed of boat = unknown

R = speed of river = unknown

Time boat continued upstream after wind blew Maria’s hat into river = 12 minutes= 12/60 hours = 0.2 hours

Distance from starting point when wind blew Maria’s hat into river = 3 km

t = Time it took for the boat to reach the starting point after it turned around to recover Maria’s hat = unknown

Boat’s distance from the starting point when it turned around to recover Maria’s hat

B_d = 0.2 * (B - R) + 3

Distance the boat traveled to pick up Maria’s hat after it turns around:

B_d = t * (B + R)

Distance the hat floated downstream after it was blown into the water

D_h = (t + 12) * R = 3


The final equations are:

B_d = 0.2 * (B - R) + 3

B_d = t * (B + R)

(t + 0.2) * R = 3


Since no further information is given, the first two equations should be combined. Since the distance is the same in both cases:

t * (B + R) = 0.2 * (B - R) + 3


The final two equations are:

t * (B + R) = 0.2 * (B - R) + 3

(t + 0.2) * R = 3

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Oct 19, 2011
querry on equation no 3
by: Anonymous

The equation no. 3 i.e (t+0.2)*R=3 seems to be confusing as time taken here (t+0.2) might be incorrect ,Because in question it is said that the time taken by HAT and by boat to reach the starting point are same. That means this equation should have been t*R=3 instead of (t+0.2)*R=3.

Oct 19, 2011
Hat on the River
by: Staff

t is only PART of the TOTAL TIME the hat drifted with the current.

t = Time it took for the boat to reach the starting point AFTER it turned around to recover Maria's hat

The total time the hat floated down the river is: t + 0.2 hours

(t+0.2)*R=3 is correct.


Thanks for writing.

Staff
www.solving-math-problems.com


Apr 20, 2017
different time taken in eqn 2 and eqn 3 NEW
by: Anonymous

If you are taking t+0.2 in eqn 3 then why you are not taking t+2 in eqn 2.. Eqn 2 was the distance when the boat turn around and you said "t" is the only part of total time when boat turn around.

Apr 20, 2017
different time taken in eqn 2 and eqn 3 NEW
by: Anonymous

If you are taking t+0.2 in eqn 3 then why you are not taking t+2 in eqn 2.. Eqn 2 was the distance when the boat turn around and you said "t" is the only part of total time when boat turn around.

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