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a. The answer is i.

b. The answer is i - sqrt(3).

c. When  7-i is converted to the exponential form re^itheta, the value of r is 4*sqrt(3).

Let's solve this problem step-by-step:

1. Define variables:

Let L be the length, W be the width, and H be the height of the rectangular prism.

Let D be the length of the diagonal joining one corner of the prism to the opposite corner.

2. Use surface area information:

The total surface area of the prism is the sum of the areas of its six faces. Using the formula for rectangular face area, we can write:

2 * L * W + 2 * W * H + 2 * L * H = 56

3. Use edge length information:

The sum of all edge lengths is equal to the perimeter of the base plus the perimeter of the lateral faces. We can write:

2 * (L + W) + 4 * H = 64

4. Simplify equations:

From equation 2, we can isolate H: H = 30 - (L + W).

Substitute this expression for H in equation 1: 2 * L * W + 2 * W * (30 - (L + W)) + 2 * L * (30 - (L + W)) = 56.

Expand and simplify: 60L - 2W^2 - 2LW = 56.

Rewrite as a quadratic equation: -2W^2 - LW + 4L - 56 = 0.

5. Solve for L and W:

We can use factoring or the quadratic formula to solve for L and W. One possible solution is L = 8 and W = 2.

6. Calculate diagonal length:

Using the Pythagorean theorem in the right triangle formed by the diagonal and the sides of the prism, we can find D: D^2 = L^2 + W^2 + H^2.

Substituting the values: D^2 = 8^2 + 2^2 + (30 - 10)^2 = 960.

Taking the square root: D = √960

Therefore, the length of the diagonal joining one corner of the prism to the opposite corner is sqrt(960) units.

Here's how to find the value of n for the equilateral triangle, square, and regular n-gon:

Analyze the pentagons and decagon:

Two pentagons and a decagon can surround a point because the angles at each vertex of the decagon can be paired with two non-consecutive angles of a pentagon, forming a straight line through the central point.

Apply similar logic to the triangle and square:

Consider the equilateral triangle. For this shape to surround a point with the square and n-gon, each vertex of the triangle must coincide with a corner of the square and a vertex of the n-gon.

Similarly, for the square, each vertex must coincide with a corner of the equilateral triangle and a vertex of the n-gon.

Connect angles:

Since each vertex of the triangle coincides with a corner of the square, the three angles at that vertex (all 60° in the triangle) must add up to 360°, the angle at the corresponding vertex of the n-gon.

Therefore, the internal angle at each vertex of the n-gon is 360° / 3 = 120°.

Identify n-gon:

A regular polygon with an internal angle of 120° is an equilateral triangle or a hexagon. However, an equilateral triangle already fills one of the spaces, so the n-gon must be a regular hexagon.

Conclusion:

Therefore, the value of n is 6. A regular hexagon, along with an equilateral triangle and a square, with all sides the same length, can perfectly surround a point, similar to the two pentagons and a decagon.

This solution demonstrates how understanding the geometric relationships between different shapes and their angles can help solve spatial reasoning problems.

In a regular polygon with n sides, we can draw nC2 diagonals (excluding the n sides themselves). Therefore, we have the equation:

nC2 = 2n

To solve for n, we can rewrite the binomial coefficient using factorials:

n(n-1) / 2 = 2n

Simplifying the equation:

n(n-1) = 4n

n^2 - n - 4n = 0

n^2 - 5n = 0

Factoring the equation:

n(n-5) = 0

Therefore, the possible values of n are 0 and 5. However, a regular polygon cannot have 0 sides. So, the only valid solution is n = 5.

Therefore, the regular polygon has 5 sides. This corresponds to a pentagon, where the number of diagonals (10) is indeed twice the number of sides (5).

Here's how to find the side length of the hexagon:

Formula for area of a regular hexagon: The area of a regular hexagon can be calculated using the formula:

A = (3√3 / 4) * s^2

where A is the area and s is the side length.

Substituting known value and solving for s: We are given that A = 3/2 square units. Plugging this into the formula:

3/2 = (3√3 / 4) * s^2

3/2 * 4/3√3 = s^2

2/√3 = s^2

Taking the square root of both sides (remembering both positive and negative solutions):

s = ± √(2/√3)

Simplifying the radical:

s = ± √(2√3 / 3)

Therefore, the possible side lengths of the hexagon are:

s1 = + √(2√3 / 3) ≈ 1.051 units (approximately)

s2 = - √(2√3 / 3) ≈ -1.051 units (negative value not applicable)

Since the side length of a polygon cannot be negative, the valid side length of the hexagon is approximately 1.051 units.

Note: This answer provides the solution in both exact and approximate forms. You can use whichever form is more appropriate for your specific context.

In a regular polygon, the sum of the interior angles is (n-2)*180 degrees, where n is the number of sides. Each interior angle of the given polygon measures 2x degrees, where x is the measure of each exterior angle.

Therefore, we have the equation:

(n-2)*180 = n * 2x

Simplifying this equation:

180n - 360 = 2nx

2x = 180 - 360/n

x = (180 - 360/n) / 2

Now, we know that the measure of each interior angle is 2x:

2x = 2 * (180 - 360/n) / 2

x = 180 - 360/n

We are given that the measure of each interior angle is twice the measure of each exterior angle, so:

2x = 2 * (180 - 360/n) = 360/n

Substituting the expression for x from before:

2 * (180 - 360/n) = 360/n

360 - 720/n = 360/n

720/n = 0

Since n cannot be zero, the only solution is if n = 720.

Therefore, the regular polygon has 720 sides.

Sure, here's how to find the slope of line AB:

Rewrite each circle equation in standard form:

x^2 + y^2 = 4 => (x^2 - 4) + y^2 = 0

(x - 5)^2 + (y - 8)^2 = 57 => (x^2 - 10x + 25) + (y^2 - 16y + 64) = 0

Add the two equations: 2x^2 - 10x + y^2 - 16y = 39

Group like terms: 2(x^2 - 5x) + y^2 - 16y = 39

Factor out the common factors: 2(x - 5)(x - 5) + y(y - 16) = 39

Solve for y: y = (39 - 2(x - 5)(x - 5)) / (y - 16)

To find the slope of line AB, we need to find the coordinates of points A and B. We can do this by substituting different values of x into the equation for y and solving. For example, if we substitute x = 5, we get y = 11. Therefore, point A is (5, 11).

Similarly, if we substitute x = 8, we get y = 0. Therefore, point B is (8, 0).

Calculate the slope of line AB: (11 - 0) / (5 - 8) = 2

Therefore, the slope of line AB is 2.