Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Perform each calculation. $$47^{\circ} 29^{\prime}-71^{\circ} 18^{\prime}$$

Step-by-Step Solution

Answer

The answer is \(-23^{\circ} 71^{\prime}\).

Step 1: Understand the Problem

We need to subtract one angle measurement from another given in degrees and minutes. The problem is stated as follows: subtract \(71^{\circ} 18^{\prime}\) from \(47^{\circ} 29^{\prime}\).

Step 2: Align the Units

We must align the units of degrees and minutes. If necessary, exchange 1 degree for 60 minutes to facilitate the subtraction.

Step 3: Subtract the Minutes

Subtract the minutes: \(29^{\prime} - 18^{\prime} = 11^{\prime}\). No borrowing is needed as \(29\) is greater than \(18\).

Step 4: Subtract the Degrees

Subtract the degrees: \(47^{\circ} - 71^{\circ}\). Since we cannot directly subtract a greater number of degrees from a smaller, we must adjust by borrowing.

Step 5: Borrow and Subtract

Borrow 1 degree from \(47^{\circ}\) and convert it into 60 minutes, adding it to the 29 minutes:\[46^{\circ} \, 89^{\prime} - 71^{\circ} \, 18^{\prime}\]Now subtract the degrees: \[46^{\circ} - 71^{\circ}\] leads to borrowing further or handle separately as shown in next step.

Step 6: Handle Negative Result

Since the result is negative as \( 46 - 71 = -25\), Interpret it as \(-24^{\circ} \) since constantly borrowing works inefficiently, redefine as \(-25^{\circ} \) then the remaining minute \(= (89-18)=71^{\prime}\) forming \(-24^{\circ} \, 71^{\prime}.\)

Step 7: Compile the Final Answer

The subtraction yields \(-23^{\circ} 71^{\prime}\) as misunderstood above due faulty then the correct final is \(-23^{\circ} 71^{\prime}\) after adjustment final seating sign.

Key Concepts

Degrees and Minutes

When calculating with angles, the units used are degrees and minutes. Understanding these units is essential because just like time, 1 degree is divided into 60 minutes. Each degree is like an hour and a minute is like a single minute of time. This means that if you ever need to convert or exchange between these units, you should always use this 1-to-60 relationship.
For angles, it looks like this:

• 1 degree (\(1^{\circ}\)) = 60 minutes (\(60^{\prime}\))

In operations, you'll often move between degrees and minutes. Knowing this will help you when you need to adjust for subtractions or handle larger calculations.
Understanding this relationship allows for accurate calculations during subtraction, ensuring leftover degrees or minutes are correctly calculated.

Borrow and Subtract

Subtracting angles can sometimes require you to "borrow," much like in basic arithmetic. When subtracting minutes, if the minuend (first number) is smaller than the subtrahend (second number), you borrow 1 degree and convert it into 60 minutes. Let's look at how borrowing works in this context:

• Suppose you have 29 minutes and need to subtract 18 minutes. Since 29 is more than 18, borrowing isn't necessary here.
• Now compare degrees. Suppose you have 47 degrees but need to subtract 71 degrees. Here, the minuend is smaller than the subtrahend, hence borrowing is essential.
• Convert 1 degree into 60 minutes and add to the minutes, making you move from 47 degrees and 29 minutes to 46 degrees and 89 minutes (29 + 60).

This adjustment allows the subtraction to be made with ease. Borrowing ensures there is enough of the smaller unit to subtract so operations don't result in negative numbers prematurely.

Negative Result Handling

Sometimes angle subtraction might result in negative outcomes. Handling negative results effectively is crucial in obtaining the accurate answer:

• If subtracting results in a negative degree, such as 46 degrees minus 71 degrees yielding -25 degrees, you need to carefully interpret this.
• By borrowing and adjusting, the negative degree can be redefined into a format more commonly used in angle measurements, like \(-24^{\circ}\) with remaining minutes adjusted correctly.
• Make sure remaining calculations like subtracting the minutes (for instance 89 - 18 = 71 minutes) are correctly done to compile the final result.

In cases where the result seems inconsistent, ensure to reevaluate the steps you took to avoid errors in the final angle measurement. Correct understanding and execution are vital to reach sound conclusions in angle subtraction. Always adjust the final result to the traditional way angles are expressed, ensuring proper sign placement and correct degree-minute relationships.