Question

The Pythagorean Sea, in Southern Antarctica has a depth that varies quadratically with its width. The width of the sea is 1000 meters. Two hundred meters from shore, the depth is 20 meters. Find the particular quadratic equation for the sea. Compute the depth of the deepest portion of the sea.

Answer 1

To find the quadratic equation that models the depth of the Pythagorean Sea, we apply given conditions to the standard form y = ax^2 + bx + c. Solving for coefficients, we then identify the vertex of the parabola to find the deepest point of the sea. Finally, we calculate the depth at this point by substituting the vertex into the equation.

Step-by-step explanation:

Finding the Particular Quadratic Equation and Depth of the Sea

To find the particular quadratic equation for the depth of the Pythagorean Sea as a function of its width, we can begin by assuming the standard form of the equation:

y = ax^2 + bx + c,

where y represents the depth in meters, and x is the horizontal distance from the shore in meters. Given that the sea's width is 1000 meters and at 200 meters from the shore the depth is 20 meters, we can establish some conditions.

• At x = 0 (shoreline), the depth is 0: hence, c = 0.
• At x = 200 meters, the depth is y = 20 meters.
• At x = 1000 meters (opposite shoreline), the depth should return to 0.

By substituting these conditions into the equation, we can solve for a and b to determine the complete equation. Once we have the equation, the depth's maximum (the deepest part of the sea) will occur at the vertex of the parabola, which is at x = -b/(2a).

Once we find the deepest point, we can calculate its depth by substituting back into the equation.

Answer 2

The depth varies quadratically with the width, so the quadratic equation has the form

d = kw^2

where d is the depth, w is the width, and k is a constant.

We can use the given information to find k.

At a width of 200 m, the depth is 20 m.

20 = k(200)^2

k = 0.0005

The equation of the depth is

d = 0.0005w^2

Now use a width of 1000 m to find the depth at the greatest width.

d = 0.0005w^2

d = 0.0005(1000)^2

d = 500

Answer: 500 m