Could you please make sure all questions are answered? thank you Sure, here's the transcription suitable for an educational website:---### Analyzing Distance Using a Quadratic FunctionSuppose the distance from home (in miles) of a car at time \( t \) hours, \( 0 \leq t \leq 8 \), is given by the quadratic function:\[ d(t) = 30t^2 - 255t + 795 \]The function is graphed below. Include units of measure in your answers below.![Graph of quadratic function: \(d(t)=30t^2-255t+795\)]The graph shows a quadratic curve that opens upwards. The x-axis represents time \( t \) in hours, ranging from 0 to 8 hours, while the y-axis represents the distance \( d(t) \) in miles. The curve starts from just below 800 miles, dips to its lowest point, and then rises again.---#### a. Finding the Average Speed**Task:**Find the average speed of the car between 3 and 7 hours.**Solution:**To find the average speed, we need to find the total distance covered between 3 and 7 hours and divide it by the total time, which is \( 7 - 3 = 4 \) hours.\[ \text{Average Speed} = \frac{d(7) - d(3)}{7 - 3} \]Substitute \( t = 3 \) and \( t = 7 \) into the function \( d(t) \):\[ d(3) = 30(3)^2 - 255(3) + 795 \]\[ d(7) = 30(7)^2 - 255(7) + 795 \]Calculate \( d(3) \) and \( d(7) \) and then the average speed.---#### b. Calculating the Minimum Distance**Task:**Calculate exactly when the car is the closest to home (a minimum distance from home). DO NOT ROUND. DO NOT USE DERIVATIVES.**Solution:**The minimum distance occurs at the vertex of the quadratic function since it opens upwards. For a quadratic function \( d(t) = at^2 + bt + c \), the time \( t \) at the vertex is given by:\[ t = -\frac{b}{2a} \]Here, \( a =

The given function is dt=30t2-255t+795 gives distance in miles when time is in hours. a. The average…

Suppose the distance from home (in miles) of a car at time t hours, 05:58 is given by the quadratic function d (t)=30x²-255x+795 The function is graphed below. Include units of measure in your answers below. dit) 1000- 900- 800 700+ 600+ 500+ 400+ 300+ 200- 100- 100+ a. Find the average speed of the car between 3 and 7 hours. b. Calculate exactly when the car the closest to home (a minimum distance from home). DO NOT ROUND. DO NOT USE DERIVATIVES. c. Suppose we restrict the domain of d(t) to 0StS3 so that d(t) is a one-to-one function. In this context of distance from home and hours traveled, write a sentence to explain the meaning of the expression d'(480)-1.5. Include numbers and units of measure in your answer. -1

Suppose the distance from home (in miles) of a car at time t hours, 05:58 is given by the quadratic function d (t)=30x²-255x+795 The function is graphed below. Include units of measure in your answers below. dit) 1000- 900- 800 700+ 600+ 500+ 400+ 300+ 200- 100- 100+ a. Find the average speed of the car between 3 and 7 hours. b. Calculate exactly when the car the closest to home (a minimum distance from home). DO NOT ROUND. DO NOT USE DERIVATIVES. c. Suppose we restrict the domain of d(t) to 0StS3 so that d(t) is a one-to-one function. In this context of distance from home and hours traveled, write a sentence to explain the meaning of the expression d'(480)-1.5. Include numbers and units of measure in your answer. -1

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Could you please make sure all questions are answered? thank you

Sure, here's the transcription suitable for an educational website:

---

### Analyzing Distance Using a Quadratic Function

Suppose the distance from home (in miles) of a car at time \( t \) hours, \( 0 \leq t \leq 8 \), is given by the quadratic function:
\[ d(t) = 30t^2 - 255t + 795 \]
The function is graphed below. Include units of measure in your answers below.

![Graph of quadratic function: \(d(t)=30t^2-255t+795\)]

The graph shows a quadratic curve that opens upwards. The x-axis represents time \( t \) in hours, ranging from 0 to 8 hours, while the y-axis represents the distance \( d(t) \) in miles. The curve starts from just below 800 miles, dips to its lowest point, and then rises again.

---

#### a. Finding the Average Speed

**Task:**
Find the average speed of the car between 3 and 7 hours.

**Solution:**
To find the average speed, we need to find the total distance covered between 3 and 7 hours and divide it by the total time, which is \( 7 - 3 = 4 \) hours.
\[ \text{Average Speed} = \frac{d(7) - d(3)}{7 - 3} \]

Substitute \( t = 3 \) and \( t = 7 \) into the function \( d(t) \):
\[ d(3) = 30(3)^2 - 255(3) + 795 \]
\[ d(7) = 30(7)^2 - 255(7) + 795 \]

Calculate \( d(3) \) and \( d(7) \) and then the average speed.

---

#### b. Calculating the Minimum Distance

**Task:**
Calculate exactly when the car is the closest to home (a minimum distance from home). DO NOT ROUND. DO NOT USE DERIVATIVES.

**Solution:**
The minimum distance occurs at the vertex of the quadratic function since it opens upwards. For a quadratic function \( d(t) = at^2 + bt + c \), the time \( t \) at the vertex is given by:
\[ t = -\frac{b}{2a} \]

Here, \( a =

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