A drinking glass 5 inches tall has a 2.5 inch diameter base. Its sides slope outward at a 4 degree angle. What is the diameter of the top of the glass?
radius bottom = 1.25
radius top = 1.25 + 5 tan 4
diameter top = 2 * radius top
Well, isn't that a tall order? Let's calculate this glassy situation, shall we?
First, let's find the circumference of the base. The formula for circumference is 2 * π * radius. Since the base has a diameter of 2.5 inches, the radius is 2.5 / 2 = 1.25 inches. Plugging that into the circumference formula, we get 2 * π * 1.25 = 7.85 inches.
Now, let's find out how much the glass widens as we move up. Using the given 4-degree angle, we can whip out some trigonometry skills. Tan(4 degrees) = opposite / adjacent, where the opposite is the height of the glass and the adjacent is the increase in width from the base to top. Rearranging the formula, we get adjacent = opposite / Tan(4 degrees).
Plugging in the height of 5 inches, we find that the increase in width is approximately 5 / Tan(4 degrees) = 71.46 inches.
Adding the increase in width to the circumference of the base, we get the circumference of the top of the glass. 7.85 inches (base circumference) + 71.46 inches (increase in width) = 79.31 inches.
Finally, we can find the diameter by dividing the circumference by π. The diameter of the top of the glass is approximately 79.31 inches / π ≈ 25.19 inches.
So, drum roll, please... the diameter of the top of the glass is approximately 25.19 inches. Now that's what I call a wide-mouthed glass! Cheers! 🍻😄
To find the diameter of the top of the glass, we need to calculate the diameter of the base and the height of the glass.
Given:
Height of the glass (h) = 5 inches
Diameter of the base (d) = 2.5 inches
Angle of the slope (θ) = 4 degrees
To find the diameter of the top (D), we can use trigonometry. Let's break down the problem into two parts:
1. Finding the height of the glass:
Since the sides of the glass slope outward, the height of the glass can be found by considering the vertical component of the angled side. We can use the tangent function:
tan(θ) = opposite / adjacent
Where:
θ = 4 degrees
opposite = height of the glass (h)
adjacent = radius of the base (r = d/2)
Rearranging the equation to solve for h:
h = tan(θ) * (d/2)
h = tan(4 degrees) * (2.5 inches / 2)
Now, calculating the value of h:
h ≈ tan(4) * (2.5 / 2)
h ≈ 0.0698 * 1.25
h ≈ 0.08725 inches (approximately)
2. Finding the diameter of the top:
The diameter of the top (D) can be calculated as follows:
D = d + 2 * h
Substituting the known values:
D = 2.5 inches + 2 * 0.08725 inches
Now, calculating the value of D:
D ≈ 2.5 + 2 * 0.08725
D ≈ 2.5 + 0.1745
D ≈ 2.6745 inches (approximately)
Therefore, the diameter of the top of the glass is approximately 2.6745 inches.
To find the diameter of the top of the glass, we need to consider the geometry of the glass and the information provided.
Here's how we can approach this problem step by step:
Step 1: Calculate the height of the glass sides.
Given that the glass is 5 inches tall, we know that the height of the glass sides is also 5 inches.
Step 2: Determine the angle at the top of the glass.
Since the sides of the glass slope outward at a 4-degree angle, we have an isosceles triangle between the base, the height, and the top of the glass. The total angle at the top of the glass will be twice the angle of the slope, so it is 2 * 4 = 8 degrees.
Step 3: Find the radius of the base.
The diameter is given as 2.5 inches, so the radius is half of that, which is 2.5 / 2 = 1.25 inches.
Step 4: Calculate the radius of the top of the glass.
To find the radius of the top, we need to consider the slope of the glass sides. As mentioned in step 2, we have an isosceles triangle. The radius at the top of the glass can be calculated using trigonometry.
First, let's find the height of the isosceles triangle, which is given by:
height = radius * tan(angle)
= 1.25 * tan(8 degrees)
Using a calculator, we find that the height is approximately 0.174 inches.
Since the height of the glass sides is 5 inches, we can now calculate the radius of the top:
radius_top = radius_base + height
= 1.25 + 0.174
= 1.424 inches
Step 5: Calculate the diameter of the top of the glass.
The diameter is simply twice the radius, so the diameter of the top of the glass is:
diameter_top = 2 * radius_top
= 2 * 1.424
= 2.848 inches
Therefore, the diameter of the top of the glass is approximately 2.848 inches.