Answer :
Let's solve this problem step-by-step.
### Step 1: Calculate the distance between the points
The endpoints of the diameter of the sphere are given as:
[tex]\[ \text{Point 1}: (4, 7, 10) \][/tex]
[tex]\[ \text{Point 2}: (-3, 5, -1) \][/tex]
The formula to calculate the distance [tex]\( d \)[/tex] between two points [tex]\((x_1, y_1, z_1)\)[/tex] and [tex]\((x_2, y_2, z_2)\)[/tex] in 3-dimensional space is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \][/tex]
By substituting the coordinates of the given points, we have:
[tex]\[ d = \sqrt{(-3 - 4)^2 + (5 - 7)^2 + (-1 - 10)^2} \][/tex]
Calculating the individual differences:
[tex]\[ \begin{align*} x_2 - x_1 & = -3 - 4 = -7 \\ y_2 - y_1 & = 5 - 7 = -2 \\ z_2 - z_1 & = -1 - 10 = -11 \end{align*} \][/tex]
Now, squaring these differences:
[tex]\[ \begin{align*} (-7)^2 & = 49 \\ (-2)^2 & = 4 \\ (-11)^2 & = 121 \end{align*} \][/tex]
Summing these squared values and taking the square root:
[tex]\[ d = \sqrt{49 + 4 + 121} = \sqrt{174} \approx 13.19090595827292 \][/tex]
Thus, the diameter of the sphere is approximately [tex]\(13.19090595827292\)[/tex].
### Step 2: Calculate the radius
The radius [tex]\( r \)[/tex] of the sphere is half of the diameter:
[tex]\[ r = \frac{d}{2} = \frac{13.19090595827292}{2} \approx 6.59545297913646 \][/tex]
Therefore, the radius of the sphere is approximately [tex]\(6.59545297913646\)[/tex].
### Step 3: Calculate the coordinates of the center of the sphere
The center of the sphere is the midpoint of the diameter's endpoints. The formula to find the midpoint [tex]\((x_m, y_m, z_m)\)[/tex] between two points [tex]\((x_1, y_1, z_1)\)[/tex] and [tex]\((x_2, y_2, z_2)\)[/tex] is:
[tex]\[ (x_m, y_m, z_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) \][/tex]
By substituting the coordinates of the given points, we have:
[tex]\[ \left(\frac{4 + (-3)}{2}, \frac{7 + 5}{2}, \frac{10 + (-1)}{2}\right) \][/tex]
Calculating the individual midpoints:
[tex]\[ \begin{align*} x_m & = \frac{4 - 3}{2} = \frac{1}{2} = 0.5 \\ y_m & = \frac{7 + 5}{2} = \frac{12}{2} = 6 \\ z_m & = \frac{10 - 1}{2} = \frac{9}{2} = 4.5 \end{align*} \][/tex]
So, the coordinates of the center of the sphere are [tex]\((0.5, 6.0, 4.5)\)[/tex].
### Summary
1. Radius of the sphere: [tex]\( \approx 6.59545297913646 \)[/tex]
2. Coordinates of the center: [tex]\( (0.5, 6.0, 4.5) \)[/tex]
These steps illustrate how the length of the radius and the coordinates of the center are determined.
### Step 1: Calculate the distance between the points
The endpoints of the diameter of the sphere are given as:
[tex]\[ \text{Point 1}: (4, 7, 10) \][/tex]
[tex]\[ \text{Point 2}: (-3, 5, -1) \][/tex]
The formula to calculate the distance [tex]\( d \)[/tex] between two points [tex]\((x_1, y_1, z_1)\)[/tex] and [tex]\((x_2, y_2, z_2)\)[/tex] in 3-dimensional space is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \][/tex]
By substituting the coordinates of the given points, we have:
[tex]\[ d = \sqrt{(-3 - 4)^2 + (5 - 7)^2 + (-1 - 10)^2} \][/tex]
Calculating the individual differences:
[tex]\[ \begin{align*} x_2 - x_1 & = -3 - 4 = -7 \\ y_2 - y_1 & = 5 - 7 = -2 \\ z_2 - z_1 & = -1 - 10 = -11 \end{align*} \][/tex]
Now, squaring these differences:
[tex]\[ \begin{align*} (-7)^2 & = 49 \\ (-2)^2 & = 4 \\ (-11)^2 & = 121 \end{align*} \][/tex]
Summing these squared values and taking the square root:
[tex]\[ d = \sqrt{49 + 4 + 121} = \sqrt{174} \approx 13.19090595827292 \][/tex]
Thus, the diameter of the sphere is approximately [tex]\(13.19090595827292\)[/tex].
### Step 2: Calculate the radius
The radius [tex]\( r \)[/tex] of the sphere is half of the diameter:
[tex]\[ r = \frac{d}{2} = \frac{13.19090595827292}{2} \approx 6.59545297913646 \][/tex]
Therefore, the radius of the sphere is approximately [tex]\(6.59545297913646\)[/tex].
### Step 3: Calculate the coordinates of the center of the sphere
The center of the sphere is the midpoint of the diameter's endpoints. The formula to find the midpoint [tex]\((x_m, y_m, z_m)\)[/tex] between two points [tex]\((x_1, y_1, z_1)\)[/tex] and [tex]\((x_2, y_2, z_2)\)[/tex] is:
[tex]\[ (x_m, y_m, z_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) \][/tex]
By substituting the coordinates of the given points, we have:
[tex]\[ \left(\frac{4 + (-3)}{2}, \frac{7 + 5}{2}, \frac{10 + (-1)}{2}\right) \][/tex]
Calculating the individual midpoints:
[tex]\[ \begin{align*} x_m & = \frac{4 - 3}{2} = \frac{1}{2} = 0.5 \\ y_m & = \frac{7 + 5}{2} = \frac{12}{2} = 6 \\ z_m & = \frac{10 - 1}{2} = \frac{9}{2} = 4.5 \end{align*} \][/tex]
So, the coordinates of the center of the sphere are [tex]\((0.5, 6.0, 4.5)\)[/tex].
### Summary
1. Radius of the sphere: [tex]\( \approx 6.59545297913646 \)[/tex]
2. Coordinates of the center: [tex]\( (0.5, 6.0, 4.5) \)[/tex]
These steps illustrate how the length of the radius and the coordinates of the center are determined.
