Javier has four cylindrical models. The heights, radii, and diagonals of the vertical cross-sections of the models are shown in the table.
A cylinder.
Model 1
radius: 14 cm
height: 48 cm
diagonal: 50 cm
Model 2
radius: 6 cm
height: 35 cm
diagonal: 37 cm
Model 3
radius: 20 cm
height: 40 cm
diagonal: 60 cm
Model 4
radius: 24 cm
height: 9 cm
diagonal: 30 cm
In which model does the lateral surface meet the base at a right angle?
Model 1
Model 2
Model 3
Model 4
In a right circular cylinder, the lateral surface meets the base at a right angle if the height and radius are perpendicular to each other.
In Model 2, the height is 35 cm and the radius is 6 cm. These dimensions are not perpendicular to each other.
In Model 3, the height is 40 cm and the radius is 20 cm. These dimensions are not perpendicular to each other.
In Model 4, the height is 9 cm and the radius is 24 cm. These dimensions are not perpendicular to each other.
Therefore, the lateral surface of Model 1 meets the base at a right angle.
Angelica calculated the distance between the two points shown on the graph below.
On a coordinate plane, line A B has points (2, negative 5) and (negative 4, 5).
Step 1: Use the point C(–4, –5) to make a right triangle with the 90 degree vertex at C.
Step 2: Determine the lengths of the legs: AC = 6 and BC = 10.
Step 3: Substitute the values into the Pythagorean theorem: 10 squared = 6 squared + c squared.
Step 4: Evaluate 10 squared = 6 squared + c squared. 100 = 36 + c squared. 64 = c squared. 8 = c.
She states the length of AB is 8 units. Which best describes the accuracy of Angelica’s solution?
Angelica is correct.
Angelica made an error determining the location of the right angle. The right angle should be at (2, 5).
Angelica made an error counting the lengths of the legs of the right triangle. The lengths should be 7 and 11.
Angelica made an error substituting the values into the Pythagorean theorem. The equation should be c squared = 6 squared + 10 squared.
Angelica made an error determining the location of the right angle. The right angle should be at (2, 5).
To find the length of AB, we need to calculate the distance between points (2, -5) and (-4, 5).
Using the distance formula, the length of AB is calculated as follows:
AB = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((-4 - 2)^2 + (5 - (-5))^2) = sqrt((-6)^2 + (10)^2) = sqrt(36 + 100) = sqrt(136).
Therefore, Angelica's solution is incorrect and the length of AB is sqrt(136) units, not 8 units.
What is the length of segment AC?
On a coordinate plane, line A C has points (3, negative 1) and (negative 5, 5).
To find the length of AC, we can use the distance formula.
The distance formula is given by:
AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
So in this case, the length of AC is:
AC = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-5 - 3)^2 + (5 - (-1))^2)
= sqrt((-8)^2 + (5 + 1)^2)
= sqrt(64 + 36)
= sqrt(100)
= 10
Therefore, the length of segment AC is 10 units.
What is the length of segment XY?
On a coordinate plane, line X Y has points (negative 4, 0) and (3, 2).
4.5 units
StartRoot 45 EndRoot units
StartRoot 53 EndRoot units
9 units
To find the length of XY, we can use the distance formula.
The distance formula is given by:
XY = sqrt((x2 - x1)^2 + (y2 - y1)^2)
So in this case, the length of XY is:
XY = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((3 - (-4))^2 + (2 - 0)^2)
= sqrt((7)^2 + (2)^2)
= sqrt(49 + 4)
= sqrt(53)
Therefore, the length of segment XY is sqrt(53) units.
Which equation correctly shows how to determine the distance between the points (9, –2) and (6, 3) on a coordinate grid?
d = StartRoot (6 minus 3) squared + (9 minus (negative 2)) squared EndRoot
d = StartRoot (6 + 3) squared + (9 + (negative 2)) squared EndRoot
d = StartRoot (6 minus 9) squared + (3 minus (negative 2)) squared EndRoot
d = StartRoot (6 + 9) squared + (3 + (negative 2)) squared EndRoot
The correct equation to determine the distance between the points (9, -2) and (6, 3) on a coordinate grid is:
d = sqrt((6 - 9)^2 + (3 - (-2))^2)
This equation represents the distance formula, which calculates the distance between two points in a Cartesian coordinate system.
To determine which model has a lateral surface that meets the base at a right angle, we need to understand the characteristics of a cylinder. In a cylinder, the lateral surface is the curved surface that wraps around the sides of the cylinder, and the base is the circular surface at the top and bottom of the cylinder.
When the lateral surface meets the base at a right angle, it means that the top and bottom surfaces are perpendicular to the curved surface. In other words, if we were to cut off the top and bottom of the cylinder, the resulting cross-section of the curved surface would meet the remaining circular base at a 90-degree angle.
To find out which model meets this criteria, we can examine the given dimensions, specifically the radii and diagonals of the vertical cross-sections. We need to find a model where the diagonal (the slanted line connecting the top and bottom of the cylinder) is a hypotenuse of a right triangle with one of the sides being the radius.
Let's examine each model:
Model 1:
- Radius: 14 cm
- Diagonal: 50 cm
To check if it meets the criteria, we can form a right triangle using the radius and half of the diagonal as the other side:
- One side = Radius = 14 cm
- Other side = Half of the Diagonal = 50 cm / 2 = 25 cm
Now, let's use the Pythagorean theorem to check if these sides form a right triangle. According to the theorem, for a right triangle, the square of the hypotenuse (diagonal in this case) should be equal to the sum of the squares of the other two sides.
Hypotenuse^2 = Side1^2 + Side2^2
50^2 = 14^2 + 25^2
2500 = 196 + 625
2500 = 821 (not true)
Since the equation is not true, Model 1 does not meet the criteria.
Now, let's check the other models using the same process:
Model 2:
- Radius: 6 cm
- Diagonal: 37 cm
Using the Pythagorean theorem:
37^2 = 6^2 + Side2^2
1369 = 36 + Side2^2
1333 = Side2^2
Since the equation is not true, Model 2 does not meet the criteria.
Model 3:
- Radius: 20 cm
- Diagonal: 60 cm
Using the Pythagorean theorem:
60^2 = 20^2 + Side2^2
3600 = 400 + Side2^2
3200 = Side2^2
Since the equation is true, Model 3 meets the criteria.
Model 4:
- Radius: 24 cm
- Diagonal: 30 cm
Using the Pythagorean theorem:
30^2 = 24^2 + Side2^2
900 = 576 + Side2^2
324 = Side2^2
Since the equation is true, Model 4 also meets the criteria.
So, the models that have the lateral surface meeting the base at a right angle are Model 3 and Model 4.