John is planning to build an arched trellis for the entrance to his botanical garden. The arch is to be in the shape of the upper half of an ellipse that is 4 feet wide at the base and 7 feet high.

Find the equation for the ellipse when a rectangular coordinate system is positioned with the origin at the center of the ellipse.

To find the equation for the ellipse in the given scenario, let's break down the problem into steps:

Step 1: Understand the shape of an ellipse
An ellipse is a set of points in a plane, the sum of whose distances from two fixed points, called foci, is constant. In this case, we are given that the arch is in the shape of the upper half of an ellipse.

Step 2: Determine the center of the ellipse
Since the origin of the coordinate system is located at the center of the ellipse, the coordinates of the center are (0, 0).

Step 3: Find the semi-major axis
The semi-major axis (a) of an ellipse represents half of the longest distance across the ellipse. In this case, the width of the base of the arch is 4 feet, so the semi-major axis can be calculated as a = 4/2 = 2 feet.

Step 4: Find the semi-minor axis
The semi-minor axis (b) of an ellipse represents half of the shortest distance across the ellipse. In this case, the height of the arch is 7 feet, so the semi-minor axis can be calculated as b = 7/2 = 3.5 feet.

Step 5: Write the equation for an ellipse
The equation for an ellipse with center at the origin is given by:
(x^2 / a^2) + (y^2 / b^2) = 1

Substituting the values of a and b we found in step 3 and 4, the equation for the ellipse becomes:
(x^2 / 2^2) + (y^2 / 3.5^2) = 1

Simplifying,
(x^2 / 4) + (y^2 / 12.25) = 1

Therefore, the equation for the ellipse, when a rectangular coordinate system is positioned with the origin at the center of the ellipse, is:
x^2 / 4 + y^2 / 12.25 = 1

To find the equation of the ellipse, we can use the standard form of the equation for an ellipse centered at the origin:

x^2/a^2 + y^2/b^2 = 1

Where "a" is the semi-major axis (half of the length of the base) and "b" is the semi-minor axis (half of the height).

In this case, since the base of the ellipse is 4 feet wide and the height is 7 feet:

a = 4/2 = 2
b = 7/2 = 3.5

So the equation for the ellipse is:

x^2/2^2 + y^2/3.5^2 = 1

Simplifying, we get:

x^2/4 + y^2/12.25 = 1

Therefore, the equation for the ellipse when a rectangular coordinate system is positioned with the origin at the center of the ellipse is:

x^2/4 + y^2/12.25 = 1

you have semi-axes of 2 and 7/2, with the longer axis vertical. So,

x^2/2^2 + y^2/(7/2)^2 = 1
or
x^2/4 + 4y^2/49 = 1