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Magazine Chapter 8: Problem 98
Dimensions of a Triangle. The height of a triangle is 4 meters longer than twice its base. Find the base and height if the area of the triangle is 10 square meters. Round to the nearest hundredth of a meter.
Short Answer
Expert verified
Base is approximately 2.32 m and height is approximately 8.64 m.
Step by step solution
01
Understand the Problem
We need to find the base and height of a triangle given that the height is 4 meters longer than twice its base, and the area of the triangle is 10 square meters.
02
Define Variables
Let the base of the triangle be \( b \) meters. Then the height can be expressed as \( h = 2b + 4 \) meters.
03
Use the Area Formula
The formula for the area of a triangle is \( A = \frac{1}{2} imes ext{base} imes ext{height} \). Substitute the known area and expressions for base and height: \( 10 = \frac{1}{2} imes b imes (2b + 4) \).
04
Simplify the Equation
Multiply both sides by 2 to eliminate the fraction: \( 20 = b(2b + 4) \). Simplify to get \( 20 = 2b^2 + 4b \).
05
Rearrange into Standard Form
Rearrange and simplify to form a quadratic equation: \( 2b^2 + 4b - 20 = 0 \).
06
Solve the Quadratic Equation
Use the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), where \( A = 2 \), \( B = 4 \), \( C = -20 \). Calculate the discriminant: \( 4^2 - 4 imes 2 imes (-20) = 16 + 160 = 176 \).
07
Calculate the Roots
The roots are \( b = \frac{-4 \pm \sqrt{176}}{4} \). Calculate \( \sqrt{176} \approx 13.27 \). So, \( b = \frac{-4 \pm 13.27}{4} \).
08
Determine the Positive Root
Since the base must be positive, choose the positive solution: \( b = \frac{-4 + 13.27}{4} \approx 2.32 \) meters.
09
Calculate the Height
Substitute \( b \approx 2.32 \) back into the expression for height: \( h = 2(2.32) + 4 = 4.64 + 4 = 8.64 \) meters.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Quadratic Equations
Solving quadratic equations is a critical skill in mathematics, especially in problems involving geometric shapes like triangles. A quadratic equation is of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. In our exercise, we derived the equation \( 2b^2 + 4b - 20 = 0 \) for the base \( b \) of a triangle. To find \( b \), we utilize the quadratic formula:\[b = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
- Identify coefficients: In this context, \( A = 2 \), \( B = 4 \), and \( C = -20 \). These are substituted into the formula.
- Calculate the discriminant: This is the part \( b^2 - 4ac \). The discriminant determines the nature of the roots. Here, \( 176 \) is positive, indicating two real roots.
- Find the roots: Substitute the values into the formula and simplify the square root \( \sqrt{176} \approx 13.27 \).
- Determine the correct solution: Since we are finding a length, only the positive root \( b = \frac{-4 + 13.27}{4} \approx 2.32 \) meters makes sense.
Area of a Triangle
The area of a triangle is a fundamental concept used in many mathematical problem-solving contexts. Calculating the area involves this simple yet crucial formula:\[A = \frac{1}{2} \times \text{base} \times \text{height}\]
- Understand the components: The base is usually the side of the triangle you select, and the height is the perpendicular distance from the base to the opposite vertex.
- Apply the formula: For our triangle, where the base was unknown and the height was expressed in terms of the base \( (h = 2b + 4) \), the formula became \( 10 = \frac{1}{2} \times b \times (2b + 4) \).
- Solve for unknowns: By substituting the values into the formula, we establish a quadratic equation that can be solved to find the length of the base.
Mathematical Problem-Solving
Tackling mathematical problems like finding the dimensions of a triangle requires a systematic approach. This involves breaking down problems, formulating equations, and logical thinking. Here's how you can approach such problems effectively:
- Identify and analyze: Clearly understand what the problem is asking. For instance, we need both the base and height of the triangle with given area constraints.
- Define variables: Assign symbols to unknowns. In this case, \( b \) for the base and expressing the height in terms of \( b \).
- Formulate relationships: Use known formulas and relationships, like the area of a triangle, to set up an equation involving the variables.
- Solve systematically: Use algebraic methods like expanding and simplifying to convert complex equations into solvable forms. Apply appropriate solutions such as the quadratic formula where necessary.
- Check and interpret: Once solutions are found, verify them and ensure they make practical sense. For instance, negative lengths wouldn't be valid in this context.
