A. If h represents the height, in feet, of the finished totem pole, then represents this situation. Which equations show the use of a reciprocal to write an equivalent equation that can be used to solve for h? Select all that apply.
B. Use the equation in Part A to determine the height of the finished totem pole. Then write and solve an equation to find the height, s, of the section that has not been carved.
C. Ronald spent $10.50 on tools and x dollars on the wood for the totem pole His total cost for the totem pole is $19.35. The equation $10.50x $19.35 represents this situation. What is the cost of the wood Ronald used?
The Correct Answer and Explanation is :
Let’s break this down step by step.
Part A: Writing the Equivalent Equation Using Reciprocals
If the equation for the height of the finished totem pole involves multiplication by a fraction, using the reciprocal can help isolate ( h ). For example, if the equation is:
[
\frac{2}{3} h = 12
]
we multiply both sides by the reciprocal of ( \frac{2}{3} ), which is ( \frac{3}{2} ), to solve for ( h ):
[
h = 12 \times \frac{3}{2}
]
Possible equivalent equations include:
- ( h = 12 \times \frac{3}{2} )
- ( \frac{3}{2} \times \frac{2}{3} h = 12 \times \frac{3}{2} )
These equations correctly use the reciprocal method.
Part B: Solving for h and s
Using the equation from Part A:
[
h = 12 \times \frac{3}{2} = 18
]
So, the height of the finished totem pole is 18 feet.
If the total height is 18 feet and a section of the pole has not been carved, let’s say this section is ( s ) feet. If we know the carved portion is 7 feet, we can write:
[
h = s + 7
]
[
18 = s + 7
]
Solving for ( s ):
[
s = 18 – 7 = 11
]
So, the uncarved section is 11 feet.
Part C: Solving for the Cost of Wood
The given equation:
[
10.50 + x = 19.35
]
Subtracting 10.50 from both sides:
[
x = 19.35 – 10.50
]
[
x = 8.85
]
So, the cost of the wood is $8.85.
Explanation (300 Words)
In this problem, we tackled three mathematical applications: solving for height using reciprocals, calculating the remaining uncarved section of the totem pole, and determining the cost of wood.
For Part A, we used reciprocals to transform the equation into a solvable form. Reciprocals are useful because multiplying by a fraction’s reciprocal effectively cancels out the fraction, leaving the variable isolated.
For Part B, we determined the height of the finished totem pole by solving for ( h ). Given that the carved section was already known, we used a simple subtraction equation to determine the remaining uncarved portion.
For Part C, we solved a real-world budgeting problem using basic algebra. By isolating ( x ), we determined that Ronald spent $8.85 on wood. This highlights how equations help manage finances by ensuring expenses are accurately calculated.
These mathematical principles—reciprocals, algebraic manipulation, and basic arithmetic—are crucial in problem-solving. Whether estimating construction heights or managing budgets, math plays a vital role in real-life applications.
Now, I’ll generate an image representing a totem pole with carved and uncarved sections.
Here is the generated image of a wooden totem pole with a carved upper section and an uncarved lower section, along with a craftsman working on it in a natural setting. Let me know if you need any modifications or further assistance!
