How do I model with equations for pre-calculus?

[davis808]

Homework Statement

I am supposed to be tutoring pre-calc, and they are currently studying modeling with equations, i haven't have pre-calculus in years, and there are no sufficient examples in this book and i could use assistance helping these kids! I feel like I should know how because the problems seem fairly simple. problem example:

Find two numbers whose sum is -24 and whose product is a maximum.

I now the answer is -12,-12, but am not sure how to show the work.

Homework Equations

A rectangle is inscribed in a semicircle of radius 10. find a function that models the area A of the rectangle in terms of its height h.

The Attempt at a Solution

 

[HallsofIvy]

Homework Statement

I am supposed to be tutoring pre-calc, and they are currently studying modeling with equations, i haven't have pre-calculus in years, and there are no sufficient examples in this book and i could use assistance helping these kids! I feel like I should know how because the problems seem fairly simple. problem example:

Find two numbers whose sum is -24 and whose product is a maximum.

I now the answer is -12,-12, but am not sure how to show the work.

Think of each clause as an equation. As soon as you see "two numbers" you should think "let x be one of the numbers and y the other" (or "a and b" or "u and v", ...). In fact, I would recommend being as specific as possible: "let x be the larger of the two numbers and y the smaller". Now, "two numbers whose sum is -24" means x+ y= what? "whose product is a maximum" means you want to find the maximum value of xy.

Homework Equations

A rectangle is inscribed in a semicircle of radius 10. find a function that models the area A of the rectangle in terms of its height h.

Draw a picture. Obviously, you draw a semicircle of radius 10 with a rectangle inside it.("Inscribed" means the vertices of the rectangle all lie on the semi-circle, either on the circle itself or on the base line).
Because you are given a geometric description but want an equation, I would recommend setting up a coordinate system. Choose your coordinates so the origin is at the center of the circle, the base of the semicircle is the x-axis, and the semicircle lies above the x- axis.
What is the equation of the semicircle in that coordinate system?

Actually this could be a very complicated problem but because of the reference to "the area A in terms of its height", you are clearly expected to assume one side of the rectangle (the "width") is along the base line of the semicircle. That simplifies things a lot!

Since the base of the rectangle lies on the x-axis, and the height is measured perpendicular to that, the height is the y value. A vertex above that is on the circle. Taking y= h, use the equation of the circle to solve for x. There will be two solutions because there are two such points. Now, from that, what is the width of the rectangle in terms of h? Area= height*length for a rectangle.

The Attempt at a Solution

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[Architeuthis Dux]

To model with equations for pre-calculus, we need to understand the concept of modeling and how it applies to equations. Modeling is the process of using equations or mathematical functions to represent real-world situations or problems. In order to model with equations for pre-calculus, we need to follow these steps:

1. Understand the problem: Before we can start modeling with equations, we need to fully understand the given problem. In this case, we are given a problem that involves finding two numbers whose sum is -24 and whose product is a maximum. We are also given another problem that involves finding the area of a rectangle inscribed in a semicircle.

2. Identify the variables: In order to create an equation, we need to identify the variables involved in the problem. In the first problem, the two numbers can be represented by x and y, where x+y=-24 and xy is the product. In the second problem, the variables are the height (h) and the area (A).

3. Write the equations: Using the identified variables, we can now write equations that represent the given problem. In the first problem, we can write the equation xy=-24x, since the sum of the two numbers is -24. To find the maximum product, we can use the concept of the quadratic formula, which gives us the equation x^2+24x=0. Solving for x, we get x=-12 or x=-12. Therefore, the two numbers are -12 and -12.

In the second problem, we can use the formula for the area of a rectangle, which is A=lw. Since the rectangle is inscribed in a semicircle, the length (l) is equal to the diameter of the semicircle, which is 2r. Therefore, the equation for the area becomes A=2rh. Substituting the given radius of 10, we get A=20h.

4. Solve the equations: Using algebraic methods, we can now solve the equations to find the values of the variables. In the first problem, we found that x=-12, so substituting this value into the equation x+y=-24, we get y=-12. Therefore, the two numbers are -12 and -12.

In the second problem, we can solve for the area by substituting the given value of the radius (10) and solving for h. This gives us A=200, which means that