A. complete the table to find an interval containing the...

a. complete the table to find an interval containing the solution of the equation, b. use a graphing utility to graph both sides of the equation to estimate the solution, and c. solve the equation algebraically. Round your results to three decimal places. $$20\left(100-e^{x / 2}\right)=500$$

a. complete the table to find an interval containing the solution of the equation,
b. use a graphing utility to graph both sides of the equation to estimate the solution, and
c. solve the equation algebraically.
Round your results to three decimal places.
$$20\left(100-e^{x / 2}\right)=500$$

Key Concepts

Exponential Functions

Exponential functions involve a constant raised to a variable exponent, and they are key in many natural and applied processes. Understanding their properties, such as how they grow or decay and how they can be manipulated algebraically, is crucial when solving equations where the variable is in the exponent.

Algebraic Manipulation Using Logarithms

When solving equations that include an exponential term, logarithms are often used to 'bring down' the exponent so that the variable can be isolated. This technique leverages the inverse relationship between exponential and logarithmic functions, allowing the transformation of an exponential equation into a linear one.

Numerical Estimation Techniques

Numerical estimation, such as constructing a table of values, is a method for approximating the solution of equations. By evaluating the function at several points, one can determine an interval where the solution lies based on changes in the sign or value of the function relative to the desired output.

Graphical Analysis for Intersection Finding

Graphical analysis involves plotting the functions on both sides of an equation to visually identify the point(s) of intersection. This approach is especially useful for estimating solutions, as the point of intersection represents the value at which the two functions are equal.

Transcript

00:02 Okay, in this question, you're asked to complete a table to find the interval containing the solution of the equation, and then use a graphing utility to graph both sides of the equation so we can estimate the solution, and then we'll solve the equation algebraically.

00:19 So desmos .com allows you to easily create a table.

00:23 So what i went ahead and did is i put in the equation 20 times 100 minus e to the x over 2, and then i put a second, equation y equals 500 so that we can find the solution or estimate the solution.

00:38 So what i'm going to do is going to go ahead and click the first line.

00:41 And then i'm going to press the settings wheel that's right there.

00:44 And you can see there's a table feature.

00:47 And then i can go ahead and i could enter in 5, 6, 7, 8, and 9.

00:59 So you see that there's plenty of other values here.

01:02 We don't need, and that's okay, you just leave them there.

01:06 But this gives us an idea of what the y values would be given these particular x values.

01:15 It looks like between 5 and 9, your range of values go about 1756 down to 200.

01:22 So now we've filled out our table of values, so we have an approximate interval of where the solution might be.

01:31 Now let's look and see what happens when we set that equation equal to 500 and find a point of intersection.

01:38 Well, what we're going to have to do is zoom out, right? because y equals 500 is going to be quite far away.

01:47 So y is equal to 500 is right here.

01:52 So it looks like, let me change this.

01:55 Actually can change your window right here.

01:57 Let's go up here.

01:58 Make this 550.

02:01 So our max is 550 so we can see.

02:04 Perfect.

02:05 So it looks to me that we can approximate our point of intersection.

02:10 It looks like maybe somewhere in that 8 .5 range, maybe, something along those lines...

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