Problem 26 - Intermediate Algebra

Question

Windsurfing. The area of a triangular sail on a windsurfing board is \(42 \mathrm{ft}^{2}\). If the length of the base of the sail is 7 feet, what is its height'? Camping. The area of a trapezoid-shaped canvas flap on a tent is 110 in. \(^{2}\) If the upper base of the flap is 8 in. long, and the height is 11 in., find the length of its lower base.

Step-by-Step Solution

Answer

The height of the sail is 12 feet, and the lower base of the trapezoid is 12 inches.

Step 1: Calculate the Height of the Triangular Sail

To find the height of the triangular sail, use the formula for the area of a triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]Here, the area \(A\) is 42 ft\(^{2}\), and the base \(b\) is 7 feet. Substitute these into the formula and solve for the height \(h\):\[ 42 = \frac{1}{2} \times 7 \times h \]\[ 42 = 3.5h \]\[ h = \frac{42}{3.5} = 12 \]So, the height of the sail is 12 feet.

Step 2: Calculate the Length of the Lower Base of the Trapezoid

To find the length of the lower base of the trapezoid-shaped canvas flap, use the formula for the area of a trapezoid:\[ \text{Area} = \frac{1}{2} \times (\text{upper base} + \text{lower base}) \times \text{height} \]Here, the area is 110 in\(^{2}\), the upper base \(a\) is 8 in., and the height \(h\) is 11 in. Substitute these into the formula and solve for the lower base \(b\):\[ 110 = \frac{1}{2} \times (8 + b) \times 11 \]\[ 110 = 5.5(8 + b) \]\[ 110 = 44 + 5.5b \]\[ 66 = 5.5b \]\[ b = \frac{66}{5.5} = 12 \]So, the length of the lower base of the trapezoid is 12 inches.

Key Concepts

Triangle Area Calculation

Finding the area of a triangle is an essential skill in geometry. The formula to calculate the area is simple:

• \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

To use this formula, we need to know two things: the length of the base and the height of the triangle.
In our windsurfing sail problem, the area was given as 42 square feet, and the base was 7 feet. We had to solve for the height, which meant rearranging the formula.
This was done by isolating the height (\[ h \]) on one side of the equation. Here’s a recap of the process:

• Start with the area formula: \[ 42 = \frac{1}{2} \times 7 \times h \].
• Simplify the equation: \[ 42 = 3.5h \].
• To get \( h \), divide both sides by 3.5: \[ h = \frac{42}{3.5} \].
• This gives the height of the triangle as 12 feet.

Knowing how to rearrange and manipulate algebraic equations is crucial in solving geometry problems, allowing you to find missing dimensions.

Trapezoid Area Formula

A trapezoid is a quadrilateral with one pair of parallel sides known as the bases. Calculating its area involves more than just multiplying base times height. The formula used is:

• \[ \text{Area} = \frac{1}{2} \times (\text{upper base} + \text{lower base}) \times \text{height} \]

In the camping scenario, the trapezoid-shaped tent flap had a known area of 110 square inches, an upper base of 8 inches, and a height of 11 inches. By applying these figures to the formula, we could find the length of the unknown lower base.
This is how you'd do it:

• Substitute the known values into the formula: \[ 110 = \frac{1}{2} \times (8 + b) \times 11 \].
• Multiply out: \[ 110 = 5.5(8 + b) \].
• Simplify to find \[ 110 = 44 + 5.5b \].
• Rearrange to isolate \( b \): \[ 66 = 5.5b \].
• Solve for \( b \): \[ b = \frac{66}{5.5} = 12 \].

This technique is straightforward. It shows the power of understanding area formulas and algebra to find unknown measures.

Algebraic Equation Solving

Algebra helps translate word problems into solvable mathematical equations. This is crucial in geometry, especially when dealing with area calculations like in the triangle and trapezoid problems.
Here's a step-by-step guide on how to solve these kinds of algebraic equations:

• Identify what you need to find, such as a missing dimension.
• Write down the area formula that relates to the shape you're dealing with.
• Substitute the known values into the equation.
• Manipulate the equation to isolate the unknown variable.
• Perform arithmetic operations to solve for the unknown.

Solving equations often involves operations such as multiplication, division, addition, or subtraction.
The aim is to have one side of the equation as the unknown quantity. Remember:

• Whatever action is taken on one side of the equation, it must be reflected on the other side to maintain balance.

Practicing algebraic manipulation will make solving more complex geometry problems intuitive and efficient.