Let l be the length and w be the width. Since the perimeter is 24, we have the equation:

24 = 2l + 2w

We know that the length is 2 more than the width, so l = w + 2. Now we can substitute l with w + 2 in the perimeter equation:

24 = 2(w + 2) + 2w
24 = 2w + 4 + 2w
24 = 4w + 4

Subtracting 4 from both sides, we get:

20 = 4w

Dividing both sides by 4, we find the width:

w = 5

Now, using the equation l = w + 2, we get:

l = 5 + 2
l = 7

So, the dimensions of the rectangle are length = 7 and width = 5.

In total, Sam has 4 + 6 + 3 = 13 pens. To find the probability of choosing a blue pen at random, divide the number of blue pens by the total number of pens:

P(blue pen) = 4 blue pens / 13 total pens ≈ 0.31

The probability of choosing a blue pen at random is approximately 0.31, or 31%.

Recall that the slope (m) of a line that passes through two points (x1, y1) and (x2, y2) can be found using the formula:

m = (y2 - y1) / (x2 - x1)

Applying the formula to our points (3, 4) and (6, 10):

m = (10 - 4) / (6 - 3) = 6 / 3 = 2

The slope of the line is 2.

To find the volume V of a cylinder, we use the formula V = πr²h, where r is the radius and h is the height.

In this case, the radius is the same as that of the sphere, which is 3 cm. The height of the cylinder is double the radius, which is the diameter of the sphere. Therefore, the height is 6 cm.

So, V = π(3²)(6) = 54π

The volume of the cylinder is 54π cubic centimeters.

The area of the rectangular garden is length times width, or 12 m * 8 m = 96 m². Since the area of the sidewalk is said to be equal to the area of the garden, the area of the sidewalk is also 96 m². The sidewalk is built around the garden, creating a larger rectangle that includes the garden and the sidewalk together. Let "w" be the width of the sidewalk. So, the length of this larger rectangle becomes 12 m + 2w and the width becomes 8 m + 2w. Since the area of any rectangle is length times width, the area of the larger rectangle becomes (12 m + 2w) * (8 m + 2w). However, we know that this area must be the area of the garden (96 m²) plus the area of the sidewalk (96 m²), which is 96 m² + 96 m² = 192 m². Therefore, we have the equation: (12 m + 2w) * (8 m + 2w) = 192 m² Solving this equation for w, you'll get approximately w = 1.23 m. Hence, the width of the sidewalk around the garden is approximately 1.23 m.