Creating and Solving Equations: Tutorial

Model each word problem as an equation, solve the equation, and match the word problem to the correct value. Note that some solutions will be approximate values. Pairs 1. 20 ----> 2. 14 ----> 3. 15 ----> 4. 7.5 --->

1. The hypotenuse of a right triangle is 25 inches, and its height is 5 inches more than its base. What is the height of the triangle? 2. Find the radius (in inches) of a circle whose area is 616 square inches. 3. A telemarketer is paid a fixed amount of $15 in addition to $2 per call. If the average cost per call is $3, find the number of calls made by the telemarketer. 4. Together, two pipes can fill a tank in 3 hrs. One pipe alone can fill the tank in 5 hrs. How long does it take for the other pipe to fill the tank alone?

A forest currently has 2,000 trees of a particular species. If the number of trees increases at the rate of 10 percent annually, how many years will it take for the forest to have 2,662 trees of that species? The initial number of trees (P) is 2,000, the rate of increase (r) is 10%, and the final value (A) is 2,662. We need to find out how many years it will take for the forest to have 2,662 trees of that species. Let t be the number of years. The number of trees increases every year, so we will use (1 + r) in the exponential equation:

A = P(1 + r)^t 2,662 = 2,000(1 + 0.1)^t (because r = 10% = 10/100 = 0.1) 2,662/2,000 = 1.1^t 1.1^3 = 1.1^t 3 = t(comparing the powers of the terms on both sides of the equation). In 3 years, the forest will have 2,662 trees of that particular species. Note that we modeled the problem as an exponential equation in which the unknown value (t) is the exponent.

Quadratic Equations

A quadratic equation is one in which the highest power of the variable is 2. A quadratic equation is also referred to as an equation of second degree. It is written in the standard form ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. Some word problems can be modeled as quadratic equations. Area calculations, for example, commonly involve quadratic equations. The six-step process you used earlier to translate word problems into linear equations also works for quadratic equations. Let's solve an area problem that can be modeled as a quadratic equation. The area of a rectangular field is 48 square meters. The length of the field is 7 meters more than 3 times its width. Find the length and width of the field

Linear Equations

To create an equation to represent a word problem, follow this six-step process: Read the problem carefully, and figure out what you need to find. Analyze the problem. Set up a strategy, and assign a variable to the quantity you need to find. Write the strategy sentences as equations or inequalities (=, <, >, ≤, or ≥). Solve the equations using the correct operations (+, -, ×, ÷). Verify that you have correctly answered what you were required to find. Check your answer. Your answer must make the equation true and also must make logical sense in the given context. You've learned that a linear equation is an equation in which the power of the variable does not exceed 1. A linear equation in one variable (x) can be written in the form ax + b = 0, where a and b are constants. Some word problems can be modeled as linear equations. Let's try a sample problem. First we'll create the equation to represent the problem, and then we'll solve the equation to find the value of the unknown quantity. Martin bought a used car with an odometer that read 7,500 miles. (An odometer shows the total number of miles a vehicle has been driven). He drives his car 600 miles each month. After a few months, the odometer shows 9,900 miles. How many months has Martin been driving the car?

Match each word problem to the equation that models it. 1. x + (x + 2) + (x + 4) = 66 2. x + (x + 1) + (x + 2) = 66 3. 2[(x + 3 + 2) + (x + 2)] = 66

1. The sum of three consecutive positive even numbers is 66. The smallest number is x. Find the three numbers. 2. The sum of three consecutive positive numbers is 66. The smallest number is x. Find the three numbers. 3. The length of a rectangle is 3 inches more than its width. If the length and width are each increased by 2 inches, the perimeter of the rectangle becomes 66 inches. If x is the width of the rectangle, find its length and width.

The area of a triangle is 17.5 square meters. The height of the triangle is 3 meters less than twice its base. The base of the triangle is x meters. Complete the equation that represents this description and fill in the values for the base and height of the triangle.

The equation modeling the described triangle is 2x^2+-3x+-35=0 The base of the triangle is 5 meters. The height of the triangle is 7 meters.

You know that the area of a rectangle is given by its length multiplied by its width. The area of the rectangular field is 48 square meters. Let the width of the rectangle be w meters and its length be l meters. The length is 7 meters more than 3 times the width. We can represent this information using the equation l = 7 + 3w.

Now, let's write the equation for area of a rectangle and solve for w. We know that the area is 48, so l x w = 48 (formula for area of a rectangle) (7 + 3w)w = 48 (substituting 7 + 3w for l) 7w + 3w^2 = 48 3w^2 + 7w - 48 = 0 We have derived a quadratic equation of the form ax2 + bx + c = 0. The coefficient a is 3, b is 7, and c is -48. Now we can factor and solve the equation: (3w + 16)(w - 3) = 0 w = -16/3 or 3 The width cannot be because the width of a rectangle cannot be a negative value, so the width must be 3 meters. That means the length (l = 7 + 3w) is 7 + 3(3), or 16 meters.

Model each investment as an exponential equation, and then arrange the investments in descending order (from greatest to least) based on their values after three years.

Sequence 1. a term CD that earns 8% each year on a starting balance of $3,000 2. a money market account that yields 10% each year on an initial deposit of $2,800 3. a fixed asset currently valued at $4,600 whose value decreases at the rate of 9% each year

Exponential Equations

Sometimes, a quantity with a certain initial value increases or decreases by a multiple. Situations in which this occurs include population growth, appreciation and depreciation of assets, and growth and decay of bacteria. Such situations can be modeled by exponential equations. In an exponential equation, the exponent is a variable. For example, an exponential equation in which P is the initial value of a quantity, r is the rate of growth or decay, t is the number of times the increase or decrease takes place, and A is the final value, is written as A = P(1 ± r)t. The term (1 + r) represents an increase, and (1 − r) represents a decrease. Let's model a real-world problem as an exponential equation. We'll create and solve the equation using the same six-step process we used earlier for linear and quadratic equations.

Scenarios that involve measuring dimensions of geometrically shaped objects can be modeled using square root equations. Let's look at an example. We'll follow the same steps for translating the word problem to a square root equation as we did for the other equation types.

Steve is remodeling his home. He plans to increase the area of his square deck by 9 square feet. If the length of each side of the new square deck is 5 feet, find the length of the sides of the original square deck. Recall that all of the sides of a square are equal in length, and its area is equal to the square of the length of the side: . The length of the sides of the new deck is 5 feet. Let x be the length of the sides of the original deck. So, the area of the original deck is x2. Therefore, the area of the new deck is x2 + 9. Let's substitute the known values in the equation: (squaring both sides of the equation) . Length cannot be negative, so we consider only the positive x-value. The sides of the original garden deck were 4 feet long.

Part B Use the equation you wrote to find the number of lessons Claire took if the average cost per lesson is $110.

The equation C = 60x + 350/x represents the average cost per lesson for x lessons. The value of C is 110. Solve for x. C = 60x + 350/x 110 = 60x + 350/x 110x = 60x + 350 50x = 350 (grouping like terms and simplifying) x = 7 If the average cost per lesson is $110, Claire took 7 lessons.

After reading the problem, we know that the odometer in Martin's car initially showed 7,500 miles. Martin drives the car 600 miles each month. The car's final odometer reading is 9,900. We need to find the number of months Martin has been driving the car.

To find the unknown number of months, we need to write an equation relating the known and the unknown values and then solve the equation to determine the unknown value. We can represent the problem with this equation: final odometer reading = initial odometer reading + (mileage per month) × (number of months). Let's assign the variable m to represent the unknown quantity, which is "number of months." Substituting the known values in the above equation, we get: 9,900 = 7,500 + (600 × m). Now, we can solve the equation to find m: 9,900 - 7500 = 600 x m 2400 = 600 x m 2400 ---------- = m 600 m = 4 Martin has used the car for 4 months.

Rational and Root Equations

We can use rational equations to model a variety of scenarios, including those that involve average rate or average cost. A rational equation involves fractions in which the numerator or the denominator, or both, contains the unknown value. Let's model a word problem with a rational equation and then solve the equation to find the unknown value. Audrey is moving and rents a truck. The truck rental company charges a base rate of $20 per day plus $0.59 per mile. If Audrey's average rental cost per mile is $1.09, how many miles did she drive the truck? We can find the average rental cost by dividing the total cost by the number of miles: average cost = total cost/number of miles. Let x be the number of miles. The total rental cost is $20 + $0.59x. The average cost is $1.09. Substitute these values in the equation and solve for x: average cost = total cost/number of miles 1.09 = 20 + 0.59x/x 1.09x = 20 + 0.59x 0.5x = 20 x = 40 Audrey drove the truck 40 miles.

Let's verify our answer by solving the quadratic equation we created to model the word problem. We'll substitute the values of length and width in the formula for the area of a rectangle and check whether we get the correct area:

area of a rectangle = length × width = 16 meters × 3 meters = 48 square meters. This answer matches the area of the rectangle given in the problem, proving that the values of length and width that we found by solving the equation are correct.

Question 1 Claire enrolls in a scuba diving program. The program costs include a one-time registration fee of $75 and $60 for each lesson. In addition, Claire had to purchase scuba equipment costing $275. Use this information to complete the activity. Part A Write an equation that models the average cost per lesson (C) in terms of the number of lessons that Claire takes (x).

average cost per lesson = total cost/number of lessons total cost = cost of scuba equipment + registration fee + (60)(number of lessons) = 275 + 75 + 60(x) cost per lesson (C) = 275 + 75 + 60(x)/x C = 350 + 60x/x The equation C = 60x + 350/x models the average cost per lesson