Ralph is 3 times as old as Sara. In 6 years , Ralph will be only twice as old as Sara will be then. Find Ralph’s age now .
1 answer:
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Number of text messages Keisha sent: 16
Number of text messages Miguel sent: 21
Number of text messages Pablo sent: 32
Step-by-step explanation:
Total messages sent = 69
Let,
Messages sent by Keisha = x
Messages sent by Miguel = y
Messages sent by Pablo = z
According to given statement;
x+y+z=69 Eqn 1
y=x+5 Eqn 2
z=2x Eqn 3
Putting Eqn 2 and 3 in Eqn 1
Dividing both sides by 4;
Putting x=16 in Eqn 2
Putting x=16 in Eqn 3
Number of text messages Keisha sent: 16
Number of text messages Miguel sent: 21
Number of text messages Pablo sent: 32
Keywords: addition, linear equation
Learn more about addition at:
#LearnwithBrainly
Answer:
(a) (x - 3)² + (y + 4)² + (z - 5)² = 25
(b) (x - 3)² + (y + 4)² + (z - 5)² = 9
(c) (x - 3)² + (y + 4)² + (z - 5)² = 16
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 = 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 = 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
Therefore, the equation of the sphere when it touches the yz plane is:
(x - 3)² + (y + 4)² + (z - 5)² = 9
(b) To get the equation of the sphere when it touches the xz-plane, we do the following:
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 = 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