Solved on Jan 19, 2024

Find the remaining distance for a linear function with slope -0.95 miles/minute, given 64 miles remaining after 37 minutes of driving. Determine the remaining distance after 57 minutes of driving.

STEP 1

Assumptions
1. The remaining distance to drive is a linear function of Milan's driving time.
2. The slope of the line representing this function is $-0.95$ .
3. Milan has $64$ miles remaining after $37$ minutes of driving.
4. We want to find the remaining distance after $57$ minutes of driving.

STEP 2

The general form of a linear function is:
$y = mx + b$
where $y$ represents the remaining distance, $m$ is the slope, $x$ is the driving time, and $b$ is the y-intercept.

STEP 3

We know the slope $m$ of the line is $-0.95$ , so we can write the function as:
$y = -0.95x + b$

STEP 4

We need to find the y-intercept $b$ . To do this, we can use the point $(37, 64)$ , where $37$ is the driving time in minutes and $64$ is the remaining distance in miles.

STEP 5

Substitute $x = 37$ and $y = 64$ into the equation to find $b$ :
$64 = -0.95 \cdot 37 + b$

STEP 6

Calculate the product of $-0.95$ and $37$ :
$-0.95 \cdot 37 = -35.15$

STEP 7

Add $35.15$ to both sides of the equation to isolate $b$ :
$64 + 35.15 = b$

STEP 8

Calculate the value of $b$ :
$b = 64 + 35.15 = 99.15$

STEP 9

Now we can write the complete linear function for the remaining distance $y$ as a function of driving time $x$ :
$y = -0.95x + 99.15$

STEP 10

To find the remaining distance after $57$ minutes of driving, substitute $x = 57$ into the equation:
$y = -0.95 \cdot 57 + 99.15$

STEP 11

Calculate the product of $-0.95$ and $57$ :
$-0.95 \cdot 57 = -54.15$

STEP 12

Add $99.15$ to $-54.15$ to find the remaining distance $y$ :
$y = -54.15 + 99.15$

STEP 13

Calculate the value of $y$ :
$y = 99.15 - 54.15 = 45$
Milan will have $45$ miles remaining after $57$ minutes of driving.

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Find the remaining distance for a linear function with slope -0.95 miles/minute, given 64 miles remaining after 37 minutes of driving. Determine the remaining distance after 57 minutes of driving.

STEP 1

Assumptions
1. The remaining distance to drive is a linear function of Milan's driving time.
2. The slope of the line representing this function is $-0.95$ .
3. Milan has $64$ miles remaining after $37$ minutes of driving.
4. We want to find the remaining distance after $57$ minutes of driving.

STEP 2

The general form of a linear function is:
$y = mx + b$
where $y$ represents the remaining distance, $m$ is the slope, $x$ is the driving time, and $b$ is the y-intercept.

STEP 3

We know the slope $m$ of the line is $-0.95$ , so we can write the function as:
$y = -0.95x + b$

STEP 4

We need to find the y-intercept $b$ . To do this, we can use the point $(37, 64)$ , where $37$ is the driving time in minutes and $64$ is the remaining distance in miles.

STEP 5

Substitute $x = 37$ and $y = 64$ into the equation to find $b$ :
$64 = -0.95 \cdot 37 + b$

STEP 6

Calculate the product of $-0.95$ and $37$ :
$-0.95 \cdot 37 = -35.15$

STEP 7

Add $35.15$ to both sides of the equation to isolate $b$ :
$64 + 35.15 = b$

STEP 8

Calculate the value of $b$ :
$b = 64 + 35.15 = 99.15$

STEP 9

Now we can write the complete linear function for the remaining distance $y$ as a function of driving time $x$ :
$y = -0.95x + 99.15$

STEP 10

To find the remaining distance after $57$ minutes of driving, substitute $x = 57$ into the equation:
$y = -0.95 \cdot 57 + 99.15$

STEP 11

Calculate the product of $-0.95$ and $57$ :
$-0.95 \cdot 57 = -54.15$

STEP 12

Add $99.15$ to $-54.15$ to find the remaining distance $y$ :
$y = -54.15 + 99.15$

STEP 13

Calculate the value of $y$ :
$y = 99.15 - 54.15 = 45$
Milan will have $45$ miles remaining after $57$ minutes of driving.

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Find the remaining distance for a linear function with slope -0.95 miles/minute, given 64 miles remaining after 37 minutes of driving. Determine the remaining distance after 57 minutes of driving.

STEP 1

STEP 2

The general form of a linear function is:y=mx+b y = mx + b y=mx+bwhere yyy represents the remaining distance, mmm is the slope, xxx is the driving time, and bbb is the y-intercept.

STEP 3

We know the slope mmm of the line is −0.95-0.95−0.95, so we can write the function as:y=−0.95x+b y = -0.95x + b y=−0.95x+b

STEP 4

We need to find the y-intercept bbb. To do this, we can use the point (37,64)(37, 64)(37,64), where 373737 is the driving time in minutes and 646464 is the remaining distance in miles.

STEP 5

Substitute x=37x = 37x=37 and y=64y = 64y=64 into the equation to find bbb:64=−0.95⋅37+b 64 = -0.95 \cdot 37 + b 64=−0.95⋅37+b

STEP 6

Calculate the product of −0.95-0.95−0.95 and 373737:−0.95⋅37=−35.15 -0.95 \cdot 37 = -35.15 −0.95⋅37=−35.15

STEP 7

Add 35.1535.1535.15 to both sides of the equation to isolate bbb:64+35.15=b 64 + 35.15 = b 64+35.15=b

STEP 8

Calculate the value of bbb:b=64+35.15=99.15 b = 64 + 35.15 = 99.15 b=64+35.15=99.15

STEP 9

Now we can write the complete linear function for the remaining distance yyy as a function of driving time xxx:y=−0.95x+99.15 y = -0.95x + 99.15 y=−0.95x+99.15

STEP 10

To find the remaining distance after 575757 minutes of driving, substitute x=57x = 57x=57 into the equation:y=−0.95⋅57+99.15 y = -0.95 \cdot 57 + 99.15 y=−0.95⋅57+99.15

STEP 11

Calculate the product of −0.95-0.95−0.95 and 575757:−0.95⋅57=−54.15 -0.95 \cdot 57 = -54.15 −0.95⋅57=−54.15

STEP 12

Add 99.1599.1599.15 to −54.15-54.15−54.15 to find the remaining distance yyy:y=−54.15+99.15 y = -54.15 + 99.15 y=−54.15+99.15

STEP 13

Calculate the value of yyy:y=99.15−54.15=45 y = 99.15 - 54.15 = 45 y=99.15−54.15=45Milan will have 454545 miles remaining after 575757 minutes of driving.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

The general form of a linear function is:
$y = mx + b$
where $y$ represents the remaining distance, $m$ is the slope, $x$ is the driving time, and $b$ is the y-intercept.

We know the slope $m$ of the line is $-0.95$ , so we can write the function as:
$y = -0.95x + b$

We need to find the y-intercept $b$ . To do this, we can use the point $(37, 64)$ , where $37$ is the driving time in minutes and $64$ is the remaining distance in miles.

Substitute $x = 37$ and $y = 64$ into the equation to find $b$ :
$64 = -0.95 \cdot 37 + b$

Calculate the product of $-0.95$ and $37$ :
$-0.95 \cdot 37 = -35.15$

Add $35.15$ to both sides of the equation to isolate $b$ :
$64 + 35.15 = b$

Calculate the value of $b$ :
$b = 64 + 35.15 = 99.15$

Now we can write the complete linear function for the remaining distance $y$ as a function of driving time $x$ :
$y = -0.95x + 99.15$

To find the remaining distance after $57$ minutes of driving, substitute $x = 57$ into the equation:
$y = -0.95 \cdot 57 + 99.15$

Calculate the product of $-0.95$ and $57$ :
$-0.95 \cdot 57 = -54.15$

Add $99.15$ to $-54.15$ to find the remaining distance $y$ :
$y = -54.15 + 99.15$

Calculate the value of $y$ :
$y = 99.15 - 54.15 = 45$
Milan will have $45$ miles remaining after $57$ minutes of driving.