The total time taken by a boat to cover 400 km downstream and 320 upstream is 40 hours. If downstream speed of the boat is 4 km/hr more than upstream speed of the boat, then find the time taken (in hours) by the boat to cover 720 km downstream.

Correct Option: 36

Let us solve the problem step-by-step.

Step 1: Define the variables for downstream and upstream speeds.
Let the upstream speed of the boat be $U$ km/hr.
According to the problem, the downstream speed is 4 km/hr more than the upstream speed.
Therefore, the downstream speed of the boat, $D$, is:
$D=U+4$

Step 2: Set up the equation using the total time given.
The total time taken to cover 400 km downstream and 320 km upstream is 40 hours.
Using the formula $\text{Time}=\frac{\text{Distance}}{\text{Speed}}$, we can write:
$\frac{400}{D}+\frac{320}{U}=40$

Substitute $D=U+4$ into the equation:
$\frac{400}{U+4}+\frac{320}{U}=40$

Step 3: Simplify and solve the quadratic equation.
Divide the entire equation by 40 to simplify the calculations:
$\frac{10}{U+4}+\frac{8}{U}=1$

Multiply through by the common denominator $U(U+4)$:
$10U+8(U+4)=U(U+4)$
$10U+8U+32=U^2+4U$
$18U+32=U^2+4U$

Rearrange the terms to form a standard quadratic equation:
$U^2-14U-32=0$

Factorize the quadratic equation:
$U^2-16U+2U-32=0$
$U(U-16)+2(U-16)=0$
$(U-16)(U+2)=0$

This gives two possible values for speed:
$U=16$ or $U=-2$
Since speed cannot be negative, we have:
$U=16\text{km/hr}$

Step 4: Find the downstream speed.
Using the relation established in Step 1:
$D=U+4=16+4=20\text{km/hr}$

Step 5: Calculate the time taken to cover 720 km downstream.
$\text{Time}=\frac{\text{Distance}}{\text{Downstream Speed}}=\frac{720}{20}=36\text{hours}$

Thus, the time taken by the boat to cover 720 km downstream is 36 hours.