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Chapter 8: Problem 16

Solve by graphing. Round to the nearest ten-thousandth. $$ 1.5^{x}=356 $$

Short Answer

Expert verified
After following the steps, you will find an x-intercept that represents the solution to this equation which can be rounded to the nearest ten-thousandth. Please note that the answer may vary depending on the precision of your graphing tool.

Step by step solution

01

Understand the problem

Transform the given equation \(1.5^{x}=356\) into a function \(y = 1.5^{x} - 356\). This allows us to graph the function in order to solve for \(x\).
02

Graph the function

Plot the function \(y = 1.5^{x} - 356\) on the graph. You may use a graphing calculator or online graphing tools and make sure to set a significant range for \(x\) values to ensure the \(x\)-intercept is visible.
03

Identify the x-intercept

After graphing the function, identify the \(x\)-intercept(s). The \(x\)-intercept is the point where the graph intersects the x-axis. That is when \(y = 0\). In this problem, the \(x\)-intercept represents the solution of the equation.
04

Round to the nearest ten-thousandth

Once you have found the \(x\)-intercept(s), round to the nearest ten-thousandth as instructed by the exercise. This will be the final answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing
Graphing is a visual way to solve equations, especially when dealing with exponential equations like \(1.5^x = 356\). This method involves representing the equation as a function and then plotting it on a coordinate plane. When graphing \(y = 1.5^x - 356\), the goal is to determine where the graph intersects the x-axis, known as the x-intercept. A graphing calculator or an online graphing tool can simplify this process by providing a clear representation of the function. By observing the graph, we can visually estimate the solution to the equation.
X-intercepts
An x-intercept is a vital concept when graphing functions. It is the point where the graph crosses the x-axis, which occurs when the function's output is zero \((y=0)\). In the context of the equation \(1.5^x = 356\), transforming it into a function \(y = 1.5^x - 356\), means finding the x-intercept will give us the value of \(x\) that makes the equation equal to 356. Essentially, solving for the x-intercept finds our solution. In graphing, the intercept is considered the solution point where the initial condition of the equation is satisfied.
Rounding
Rounding is an essential mathematical technique for approximating numbers. In the case of finding the x-intercept from a graph, precision is crucial, especially when instructed to round to the nearest ten-thousandth. If the x-intercept is not a precise whole number, rounding helps provide a sensible approximation. The rule for rounding to the ten-thousandth place involves looking at the fifth decimal place; if it's five or higher, you increase the fourth decimal place by one, otherwise, you leave it as it is. This accurate rounding offers a clear, concise, and usable solution.
Functions
Functions form the backbone of equations and graphing. In mathematics, a function expresses a particular relationship between a set of inputs and a set of allowable outputs with a rule of correspondence. Here, the given equation \(1.5^x = 356\) is transformed into a function \(y = 1.5^x - 356\), to facilitate graphing and finding the solution. This representation allows us to utilize function properties, making it easier to analyze and solve complex problems through visual tools. Once plotted as a graph, functions can reveal critical information about the behavior of the equation.

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