a drinking glass 5 inches tall has a 2.5 inch diameter base. its sides slope upward at a 4 degree angle. what is the diameter of the top of the glass?

2.5 + 2 * 5 * tan 4

= 2.5 + 10(.07)
= 3.2 inches

To find the diameter of the top of the glass, we can start by calculating the height of the cone-shaped section of the glass.

Given:
Height of the glass (h) = 5 inches
Diameter of the base (d1) = 2.5 inches
Angle of the sides (θ) = 4 degrees

To calculate the height of the cone section, we can use the formula:

h_cone = tan(θ) * r_cone

Where r_cone is the radius of the cone section.

The radius of the base (r1) can be calculated by dividing the diameter (d1) by 2:

r1 = d1 / 2

Substituting the given values, we have:

r1 = 2.5 inches / 2
r1 = 1.25 inches

Now we can calculate the height of the cone section:

h_cone = tan(4 degrees) * 1.25 inches

Using a scientific calculator, we find that tan(4 degrees) is approximately 0.0698.

h_cone = 0.0698 * 1.25 inches
h_cone ≈ 0.08725 inches

Now, we can find the radius of the top section of the glass (r2) by subtracting the height of the cone section (h_cone) from the total height of the glass (h):

r2 = r1 - h_cone

r2 = 1.25 inches - 0.08725 inches
r2 ≈ 1.16275 inches

Finally, the diameter of the top of the glass is twice the radius:

d2 = 2 * r2

d2 = 2 * 1.16275 inches
d2 ≈ 2.3255 inches

Therefore, the diameter of the top of the glass is approximately 2.3255 inches.

To calculate the diameter of the top of the glass, we need to determine the height of the glass where the top begins, using geometric principles.

Step 1: Calculate the height where the top begins
Since the glass has a 4-degree angle, we can imagine a right triangle formed by the height, the base (diameter of the bottom), and the side of the glass. The formula for finding the height in this triangle is:

Height = tan(angle) * base

In this case, the angle is 4 degrees, and the base is given as 2.5 inches. Therefore:

Height = tan(4 degrees) * 2.5 inches

Using a scientific calculator, we can find that tan(4 degrees) is approximately 0.069756, so the height is:

Height = 0.069756 * 2.5 inches
Height = 0.17439 inches (approximately)

Step 2: Calculate the diameter of the top
Since the sides of the glass slope upward, the diameter at the top will be smaller than the diameter at the base. To find the diameter at the top, we need to subtract twice the height from the diameter at the base:

Diameter of top = Diameter of base - 2 * Height

The diameter at the base is given as 2.5 inches, and the height has been calculated as 0.17439 inches. Plugging in these values:

Diameter of top = 2.5 inches - 2 * 0.17439 inches
Diameter of top ≈ 2.15122 inches (approximately)

Therefore, the diameter of the top of the glass is approximately 2.15122 inches.

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