Math Problem Statement

What is the length of the side labeled 'x' in each diagram? Sketch the graph of each polynomial function and identify all zeros. Solve the logarithmic and exponential equations provided.

Solution

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8.117

What is the length of the side labeled "x" in each diagram?
a. A right triangle with a 60° angle and hypotenuse = 2.
b. A right triangle with a 45° angle and one side labeled "5√2".


8.119

Sketch the graph of each polynomial function below and identify all of the zeros.
a. y=x3+1y = x^3 + 1
b. y=x28y = x^2 - 8


8.124

Due to the worsened economy, mechanics in downtown Hollywood cannot afford to replace their outdoor light bulbs when they burn out. On average, about 1.8% of the lights burn out every month. Assuming there are now 1,000,000 outdoor lights in Hollywood, how long will it take until there are only 100,000 lights left?


8.125

Rewrite each expression as a single logarithm.
a. log(3y)+log(2x)\log(3y) + \log(2x)
b. log(15m)log(k)+log(n)\log(15m) - \log(k) + \log(n)


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Math Problem Analysis

Mathematical Concepts

Trigonometry
Polynomials
Logarithms
Exponential Decay

Formulas

Right triangle trigonometric ratios
Polynomial root finding
Logarithmic properties: log(a) + log(b) = log(ab), log(a) - log(b) = log(a/b)
Exponential decay formula: N(t) = N_0 * e^(-rt)

Theorems

Pythagorean theorem
Fundamental theorem of algebra
Properties of logarithms

Suitable Grade Level

Grades 10-12