booboo44

The answer is 5600 milliamps of current.

The solution is (s,t) = (-4/11, 5/11).

1. The answer is a = 15.

2. The answer is a = -27.

Let's denote:

P = number of professors

L = number of lecturers

We know:

P + L = 228 (total faculty members)

P = 5 * L (5 professors for every 14 lecturers)

Substituting the second equation into the first:

5 * L + L = 228

6 * L = 228

L = 228 / 6

L = 38

Now, we can find the number of professors:

P = 5 * L

P = 5 * 38

P = 190

Therefore, the community college employs:

190 professors

38 lecturers

In total, they employ 228 faculty members.

Analyzing the Problem

Understanding the Geometry:

We have two concentric circles (circles with the same center).

A chord of the larger circle is tangent to the smaller circle. This means the chord is perpendicular to the radius of the smaller circle at the point of tangency.

We need to find the area of the ring-shaped region, which is the difference in areas between the larger and smaller circles.

Key Points:

The chord of the larger circle is the diameter of the smaller circle.

The radius of the smaller circle is half the length of the chord.

Solution

Find the radius of the smaller circle:

Since the chord is the diameter, the radius of the smaller circle is 10/2 = 5 units.

Find the area of the smaller circle:

Area of a circle = πr², where r is the radius.

Area of the smaller circle = π(5)² = 25π square units.

Find the radius of the larger circle:

The radius of the larger circle is the sum of the radius of the smaller circle and the distance from the center to the chord (which is the same as the radius of the smaller circle).

Radius of the larger circle = 5 + 5 = 10 units.

Find the area of the larger circle:

Area of the larger circle = π(10)² = 100π square units.

Find the area of the ring-shaped region:

Area of the ring-shaped region = Area of the larger circle - Area of the smaller circle.

Area of the ring-shaped region = 100π - 25π = 75π square units.

Therefore, the area of the ring-shaped region is 75π square units.

The answer is 0.

Analyzing f(x)⋅g(x)

We are asked to find several properties of the product of two polynomials, $f(x) = -x^2+ 8x - 5$ and $g(x) = x^3 - 11x^2 + 2x$.

Finding the Product:

The product f(x)⋅g(x) can be found using the distributive property or polynomial multiplication techniques. Here, we'll use the distributive property:

f(x) * g(x) = (-x^2 + 8x - 5) * (x^3 - 11x^2 + 2x)

Multiplying each term of f(x) by each term of g(x) and combining like terms will result in a fourth-degree polynomial.

Properties of the Product:

Leading Term:

The leading term in a polynomial is the term with the highest degree. In the product, the highest degree term will come from multiplying the highest degree terms of f(x) and g(x). These terms are -x^2 and x^3, respectively. Their product is -x^5. Therefore, the leading term of f(x) * g(x) is -x^5.

Leading Coefficient:

The leading coefficient is the coefficient of the leading term. In this case, the leading term is -x^5, and its coefficient is -1. Therefore, the leading coefficient is -1.

Degree:

The degree of a polynomial is the highest exponent of the variable. Since the leading term has a degree of 5, the degree of f(x) * g(x) is 5.

Constant Term:

The constant term is the term that doesn't include any variable (x). To find it, we identify the terms in the product that don't have any x. Multiplying the constant term of f(x) (-5) by the constant term of g(x) (2) will result in the constant term of the product. Additionally, there might be other terms that don't have x due to the product of other terms. After multiplying and combining like terms, you'll find the constant term.

Coefficient of x^2:

The coefficient of x^2 is the number multiplying the term with x^2. This can be found by identifying terms in the product that have x^2. Multiply terms in f(x) that contain x by terms in g(x) that contain x^2, and vice versa. Combine like terms to find the coefficient of x^2.

Finding these properties requires multiplying f(x) and g(x) and performing some term manipulation. However, the steps outlined above should guide you in finding each property.