Question

The town of Woodgreen offers billboard space along the highway. A 5 foot by 8 foot rectangular advertising space costs $$\$ 140$$. The price \((p)\) of a sign is proportional to its area. A new sign erected in the billboard space costs $$\$ 336$$. If the new sign is 8 feet tall, then what is its length? A. 11 feet B. 12 feet C. 16 feet D. 42 feet

Short answer

The short version of the answer is: First, find the area of the 5x8 sign: \(A = 5 \times 8 = 40 \text{ sq feet}\). Next, calculate the proportionality constant: \(k = \frac{140}{(5 \times 8)} = \frac{140}{40} = 3.5\) Then, find the area of the new sign: \(A_\text{new} = \frac{336}{k} = \frac{336}{3.5} = 96 \text{ sq feet}\) Finally, calculate the length of the new sign: \[8 \times L_\text{new} = 96\] which gives us \(L_\text{new} = 12 \text{ feet}\). Hence, the answer is: B. 12 feet

Step-by-step solution

Determine the area of the 5x8 sign

The given sign has dimensions of 5 feet by 8 feet. To find its area, multiply the length by the width: \(A = 5 \times 8\).

Calculate the proportionality constant

The price is proportional to the area of the sign, meaning that: \[p = k \times A\] where p is the price, k is the proportionality constant and A is the area of the sign. Using the given information, the cost of a 5 foot by 8-foot sign is $140. So, \[140 = k \times (5 \times 8)\] Solve for k: \(k = \frac{140}{(5 \times 8)}\)

Find the area of the new sign

Now that we have the proportionality constant, we can determine the area of the new sign using its price of $336. Using the equation for the price, we have: \[336 = k \times A_\text{new}\] Plug in the value of k from Step 2 and solve for \(A_\text{new}\): \[A_\text{new} = \frac{336}{k}\]

Calculate the length of the new sign

We know the height of the new sign is 8 feet. If we want to find the length, we can set up the equation: \[A_\text{new} = 8 \times L_\text{new}\] where \(L_\text{new}\) is the length of the new sign. Substitute the value of \(A_\text{new}\) obtained in Step 3 and solve for \(L_\text{new}\): \[8 \times L_\text{new} = \frac{336}{k}\]

Find the length of the new sign

After solving for \(L_\text{new}\) in the previous step, compare the value obtained to the options given to determine the correct answer. A. 11 feet B. 12 feet C. 16 feet D. 42 feet

Key concepts

Understanding Proportional Relationships

In mathematics, proportional relationships are a cornerstone concept that links two quantities in a way that their ratio remains constant. When we say that one quantity is proportional to another, it means that if one quantity changes, the other changes at a consistent rate. This consistency is what we call the 'constant of proportionality', denoted typically by the letter 'k'.

For instance, let's say you're buying fruits, and the price of fruits is directly proportional to their weight. If 2 pounds of fruit cost \(6, then 4 pounds (which is double the weight) would cost \)12, and so on. The price per weight unit (pound) stays the same, illustrating a proportional relationship. In our GED Math Practice problem, the price of a billboard sign is proportional to its area. Therefore, we can write the equation for the price as: \( p = k \times A \) where \( p \) is the price, \( k \) is the constant of proportionality, and \( A \) is the area of the sign.

In real-life problems like these, understanding proportional relationships helps in predicting costs, quantities, and other factors with simplicity and accuracy.

Calculating the Area of a Rectangle

Speaking of area, it's a measure indicating the size of a two-dimensional surface. For a rectangle, the area is calculated by multiplying the length and the width of the rectangle. The formula for the area (\( A \)) of a rectangle is: \[ A = \text{length} \times \text{width} \]
Using the dimensions provided, if we take a rectangle with a length of 5 feet and a width of 8 feet, we simply multiply these two numbers together to find the area: \( A = 5 \text{ ft} \times 8 \text{ ft} \).

Knowing how to calculate the area is essential for various real-life scenarios, such as determining the amount of paint needed for a wall, the size of a carpet for your room, or, like our exercise, the cost of a billboard sign by its size.

Solving for Variables

In algebra, solving for variables is the process of finding the values of the unknowns in equations. To solve for a variable, we often use algebraic manipulation, which involves methods like addition, subtraction, division, and multiplication, alongside the properties of equality to isolate the variable and find its value.

In our practice problem, after finding the proportionality constant, we need to solve for the length of the new sign (\( L_\text{new} \)). By using the formula for the area of a rectangle (\( A = \text{length} \times \text{width} \)), and knowing that in this case, the area is equal to the cost divided by the constant of proportionality (\( A_\text{new} = \frac{p}{k} \)), we can rearrange the equation to isolate and solve for the length (\( L_\text{new} \)). The steps of such algebraic manipulation are fundamental and highly applicable in various disciplines within and beyond mathematics.