1. Maximum and minimum problems.

1. A man in a row boat at point P, 5 km from the nearest point A on a straight shore wishes to reasvch point B, 6 km along the shore from point A, in the shortest time. Where should he land if he can row 2km/hr and walk 4km/hr (Hint: pick a point on the shore line)?

2. An open rectangular tank (no top) with square ends is to have a volume of 6400 cubic meters. The base costs $75 per square meter and the sides$25 per square meter. What are the dimensions to build the tank at minimum cost?

2. Max and min

Hello iz1hp
Originally Posted by iz1hp
1. A man in a row boat at point P, 5 km from the nearest point A on a straight shore wishes to reasvch point B, 6 km along the shore from point A, in the shortest time. Where should he land if he can row 2km/hr and walk 4km/hr (Hint: pick a point on the shore line)?
Suppose he lands at a point T, where $AT = x$ km. Then, by Pythagoras, $PT^2 = 25 + x^2$

$\Rightarrow PT = \sqrt{25+x^2}$ km

So this part of the journey takes ...?... hours.

$BT =$ ...?... km

So this part takes ...?... hours.

Add the two times together, to find the total time $t$. Then find the value of $x$ that makes $t$ a minimum.

(Answer: $x = \frac{5}{\sqrt{3}}= 2.89$ km from A)
2. An open rectangular tank (no top) with square ends is to have a volume of 6400 cubic meters. The base costs $75 per square meter and the sides$25 per square meter. What are the dimensions to build the tank at minimum cost?
The ends of the tank are squares. Let's suppose they measure $x$ meters by $x$ meters. And suppose that the tank is $y$ meters long. Then we have:

Area of base = $xy\, m^2$, at $75 per $m^2$. So the cost of the base =$...?...

The total area of all four sides is $(2xy + 2x^2)\, m^2$ at $25 per $m^2$. So the cost of the sides =$...?...

So the total cost \$C = ...?... (in terms of $x$ and $y$)

Now the volume of the tank = area of base x height = $xy \times x = x^2y \,m^3$.

But this is $6400 \,m^3$. So $x^2y = 6400$. So $y =$ ...?... (in terms of $x$).

Now express $C$ in terms of $x$ only, and find the value of $x$ that will make $C$ a minimum.

(Answer: the tank measures 20 x 20 x 16 m.)

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a man in a boat 6 km from the nearest point b

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