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polet [3.4K]
3 years ago
9

A train from Portland, Oregon, to Los Angeles, California, travels at an average speed of 60 miles per hour and covers a distanc

e of 963 miles. Susana is taking the train from Portland to Los Angeles to see her aunt. She needs to arrive at her aunts' house by 8 p.M. It takes 30 minuets to get from the train station to her aunt's house. What time does the train need to leave Portland for Susana to arrive by 8 p.M.
Mathematics
1 answer:
seropon [69]3 years ago
8 0

Answer:

The train need to leave Portland at 03:27 am

Step-by-step explanation:

step 1

Find out how long it takes the train to travel from Portland, Oregon, to Los Angeles, California

Remember that

The speed is equal to divide the distance by the time

so

The time is equal to divide the distance by the speed

Let

s ---> the speed in miles per hour

d ---> the distance in miles

t ---> the time in hours

t=\frac{d}{s}

we have

d=963\ mi

s=60\ mi/h

substitute

t=\frac{963}{60}=16.05\ hours

step 2

Adds 30 minutes (time it takes to get from the train station to her aunt's house)

Remember that

30\ minutes=0.50\ hours

16.05+0.50=16.55\ hours

Convert to minutes

16.55\ h=16.55(60)\ min=993\ min

step 3

Remember that

8 p.m=20:00\ hrs

Convert to minutes

20(60)=1,200\ min

Subtract 993 minutes from 1,200 minutes

1,200-993=207\ min

Convert to hours+minutes

207\ min=207/60=3.45\ hours

0.45\ h=0.45(60)=27\ min

so

3\ h+27\ min

00:00 + 03:27=03:27\ am

therefore

The train need to leave Portland at 03:27 am

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Step-by-step explanation:

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3 years ago
Find equations of the spheres with center(3, −4, 5) that touch the following planes.a. xy-plane b. yz- plane c. xz-plane
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Answer:

(a) (x - 3)² + (y + 4)² + (z - 5)² = 25

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Step-by-step explanation:

The equation of a sphere is given by:

(x - x₀)² + (y - y₀)² + (z - z₀)² = r²            ---------------(i)

Where;

(x₀, y₀, z₀) is the center of the sphere

r is the radius of the sphere

Given:

Sphere centered at (3, -4, 5)

=> (x₀, y₀, z₀) = (3, -4, 5)

(a) To get the equation of the sphere when it touches the xy-plane, we do the following:

i.  Since the sphere touches the xy-plane, it means the z-component of its centre is 0.

Therefore, we have the sphere now centered at (3, -4, 0).

Using the distance formula, we can get the distance d, between the initial points (3, -4, 5) and the new points (3, -4, 0) as follows;

d = \sqrt{(3-3)^2+ (-4 - (-4))^2 + (0-5)^2}

d = \sqrt{(3-3)^2+ (-4 + 4)^2 + (0-5)^2}

d = \sqrt{(0)^2+ (0)^2 + (-5)^2}

d = \sqrt{(25)}

d = 5

This distance is the radius of the sphere at that point. i.e r = 5

Now substitute this value r = 5 into the general equation of a sphere given in equation (i) above as follows;

(x - 3)² + (y - (-4))² + (z - 5)² = 5²  

(x - 3)² + (y + 4)² + (z - 5)² = 25  

Therefore, the equation of the sphere when it touches the xy plane is:

(x - 3)² + (y + 4)² + (z - 5)² = 25  

(b) To get the equation of the sphere when it touches the yz-plane, we do the following:

i.  Since the sphere touches the yz-plane, it means the x-component of its centre is 0.

Therefore, we have the sphere now centered at (0, -4, 5).

Using the distance formula, we can get the distance d, between the initial points (3, -4, 5) and the new points (0, -4, 5) as follows;

d = \sqrt{(0-3)^2+ (-4 - (-4))^2 + (5-5)^2}

d = \sqrt{(-3)^2+ (-4 + 4)^2 + (5-5)^2}

d = \sqrt{(-3)^2 + (0)^2+ (0)^2}

d = \sqrt{(9)}

d = 3

This distance is the radius of the sphere at that point. i.e r = 3

Now substitute this value r = 3 into the general equation of a sphere given in equation (i) above as follows;

(x - 3)² + (y - (-4))² + (z - 5)² = 3²  

(x - 3)² + (y + 4)² + (z - 5)² = 9  

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i.  Since the sphere touches the xz-plane, it means the y-component of its centre is 0.

Therefore, we have the sphere now centered at (3, 0, 5).

Using the distance formula, we can get the distance d, between the initial points (3, -4, 5) and the new points (3, 0, 5) as follows;

d = \sqrt{(3-3)^2+ (0 - (-4))^2 + (5-5)^2}

d = \sqrt{(3-3)^2+ (0+4)^2 + (5-5)^2}

d = \sqrt{(0)^2 + (4)^2+ (0)^2}

d = \sqrt{(16)}

d = 4

This distance is the radius of the sphere at that point. i.e r = 4

Now substitute this value r = 4 into the general equation of a sphere given in equation (i) above as follows;

(x - 3)² + (y - (-4))² + (z - 5)² = 4²  

(x - 3)² + (y + 4)² + (z - 5)² = 16  

Therefore, the equation of the sphere when it touches the xz plane is:

(x - 3)² + (y + 4)² + (z - 5)² = 16

 

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Answer:

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