Question

Denny leaves on a drive across the state at 8:00 a.m. In the afternoon, his average speed in miles per hour is given by \(\frac{d}{t+4}\), where \(t\) is the number of hours past noon and \(d\) is the distance driven. Find Denny's average speed at 2:00 p.m, when he has driven 390 miles. A. 70 miles per hour B. 60 miles per hour C. 75 miles per hour D. 65 miles per hour

Short answer

D. 65 miles per hour

Step-by-step solution

Identify variables and given values

Let's identify the given information: The time of interest is 2:00 p.m and the distance driven is 390 miles.

Calculate the number of hours past noon

To find the number of hours past noon at 2:00 p.m., calculate: \[t = 2:00 \, p.m. - 12:00 \, p.m. = 2 \, hours\]

Substitute values into the speed formula

The average speed is given by the formula \[ \text{Speed} = \frac{d}{t+4} \] Substituting the given values: \[ \text{Speed} = \frac{390}{2+4} = \frac{390}{6} \]

Simplify the fraction

Now, simplify the expression: \[ \text{Speed} = \frac{390}{6} = 65 \, miles \, per \, hour \]

Select the correct answer

Find the correct answer choice corresponding to the calculated speed: D. 65 miles per hour

Key concepts

distance-time relationship

Understanding the distance-time relationship is key to solving problems related to average speed. The distance formula ties together these two concepts: distance (d) equals speed (v) multiplied by time (t), expressed as \( d = vt \). However, in many real-world problems, the formula may be rearranged or adapted. For instance, when you need to find either distance, time, or speed, you can rearrange the formula to fit the given variables. In our case, the average speed formula given used time and distance in different ways. Here, knowing the time after noon and distance allowed us to calculate the speed. Simplifying this can help solve many problems efficiently and is something you’ll often encounter.

algebraic substitution

Algebraic substitution is a powerful tool, used to replace variables with their actual values in mathematical equations. This allows you to simplify and solve equations more easily. In Denny's problem, you’re given specific values for the distance \(d\) and the hours past noon \(t\). By substituting these into the given formula for speed, you turn an abstract equation into a simple calculation. This process of substitution will help you break down complex problems into manageable steps. For example, you substituted \(d = 390\) and \(t = 2\) into the speed formula \( \text{Speed} = \frac{d}{t+4} \) to arrive at an answer. Make sure every step of substitution is correct to avoid mistakes!

problem solving steps

Breaking down a problem into clear, logical steps is crucial for understanding and solving it efficiently. Here, each step in the solution was crafted to make the process straightforward:

• Identify known values: Recognize what information is given. In this case, the distance (390 miles) and time (2 hours past noon).
• Calculate time past a specific point: Calculate hours past noon, which was 2.
• Substitute values: Plug given values into the formula. Replace \(d\) with 390 miles and \(t\) with 2 hours in \( \frac{d}{t+4} \)
• Simplify: Simplify the equation to find the speed. This results in a clear answer of 65 miles per hour.
• Select the correct answer: Match your calculated speed with the given options. This systematic process ensures you don’t miss any critical details.

simplifying fractions

Fractions can be simplified to their most basic form to make calculations easier. Simplifying is particularly useful when dealing with formulas like the speed one in Denny's problem. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD). For example:

• In the problem, we got the fraction \( \frac{390}{6} \)
• We divide both 390 and 6 by their GCD, which is 6
• This simplifies to \( 65 \), giving us 65 miles per hour.

Being able to quickly simplify fractions helps in solving not just homework problems but any future real-world calculations you might encounter. Practicing this will enhance your overall math skills and confidence.” } ]} } } } } } } } } } } } } } } } } } ] } } } } } } } } } } } } } } } } } } } } } } } } } } } } } ] } } } } } } } } } } ] } } }] } } } } } } }] } } } } } }] } } mun您system's json is missing the the a of400 missingEOF}} of missing and original input- parse a after output's , }} } of mexpectedIt's