30. a large cube has a side length of 10 meters. inside this large cube, non-overlapping, there are three smaller cubes with side lengths of 2 meters, 3 meters, and 4 meters. what is the volume inside the larger cube that is not occupied by one of the smaller cubes?
a) 901 m³
b) 919 m³
c) 983 m³
d) 999 m³

31. consider the function y = 2sin(x). which of the following functions below has half the amplitude and half the period of the original function?
a) 4sin(2x)
b) sin(2x)
c) 4sin(0.5x)
d) sin(0.5x)

32. if m = 3n + 5, and 2m = 5p + 20, what is n in terms of p?
a) n = (5p - 40) / 6
b) n = (5p + 10) / 6
c) n = (2p - 10) / 5
d) n = (5p - 10) / 3

33. a right triangle has an area of $\frac{49\sqrt{3}}{8}$ square units (approximately 10.613). its hypotenuse measures 7 units, and the sine of one of its angles is 0.5. what are the measures of its angles?
a) 30°, 60°, 90°
b) 41.8°, 48.2°, 90°
c) 36.9°, 53.1°, 90°
d) there is not enough information provided to answer the question.

34. consider the following two circles. one circle is given by the equation x² + y² = 2. a second circle is given by the equation (x-2)² + y² = 2. which of the following is a possible intersection point of the two circles?
a) (-1, -1)
b) (1, 1)
c) (2, 0)
d) (-2, 0)

35. consider a rectangular prism of dimensions 3 m by 4 m by 12 m. what is the length of the diagonal (distance from one corner of the prism to the far opposite corner)?
a) 13 m
b) 15 m
c) 17 m
d) 19.5 m

Question

30. a large cube has a side length of 10 meters. inside this large cube, non-overlapping, there are three smaller cubes with side lengths of 2 meters, 3 meters, and 4 meters. what is the volume inside the larger cube that is not occupied by one of the smaller cubes?
a) 901 m³
b) 919 m³
c) 983 m³
d) 999 m³

31. consider the function y = 2sin(x). which of the following functions below has half the amplitude and half the period of the original function?
a) 4sin(2x)
b) sin(2x)
c) 4sin(0.5x)
d) sin(0.5x)

32. if m = 3n + 5, and 2m = 5p + 20, what is n in terms of p?
a) n = (5p - 40) / 6
b) n = (5p + 10) / 6
c) n = (2p - 10) / 5
d) n = (5p - 10) / 3

33. a right triangle has an area of $\frac{49\sqrt{3}}{8}$ square units (approximately 10.613). its hypotenuse measures 7 units, and the sine of one of its angles is 0.5. what are the measures of its angles?
a) 30°, 60°, 90°
b) 41.8°, 48.2°, 90°
c) 36.9°, 53.1°, 90°
d) there is not enough information provided to answer the question.

34. consider the following two circles. one circle is given by the equation x² + y² = 2. a second circle is given by the equation (x-2)² + y² = 2. which of the following is a possible intersection point of the two circles?
a) (-1, -1)
b) (1, 1)
c) (2, 0)
d) (-2, 0)

35. consider a rectangular prism of dimensions 3 m by 4 m by 12 m. what is the length of the diagonal (distance from one corner of the prism to the far opposite corner)?
a) 13 m
b) 15 m
c) 17 m
d) 19.5 m

Accepted Answer

### Question 30
# Explanation:
## Step1: Calculate volume of large cube
The volume of a cube is given by $ V = s^3 $, where $ s $ is the side length. For the large cube, $ s = 10 $ meters. So, $ V_{\text{large}} = 10^3 = 1000 \, \text{m}^3 $.
## Step2: Calculate volumes of small cubes
- For the cube with side length $ 2 $ meters: $ V_1 = 2^3 = 8 \, \text{m}^3 $
- For the cube with side length $ 3 $ meters: $ V_2 = 3^3 = 27 \, \text{m}^3 $
- For the cube with side length $ 4 $ meters: $ V_3 = 4^3 = 64 \, \text{m}^3 $
## Step3: Calculate total volume of small cubes
Sum the volumes of the three small cubes: $ V_{\text{small total}} = 8 + 27 + 64 = 99 \, \text{m}^3 $
## Step4: Calculate unoccupied volume
Subtract the total volume of small cubes from the volume of the large cube: $ V_{\text{unoccupied}} = 1000 - 99 = 901 \, \text{m}^3 $
# Answer:
A) $ 901 \, \text{m}^3 $

### Question 31
# Explanation:
## Step1: Recall amplitude and period of sine function
The general form of a sine function is $ y = A\sin(Bx) $, where the amplitude is $ |A| $ and the period is $ \frac{2\pi}{|B|} $. For the original function $ y = 2\sin(x) $, amplitude $ A = 2 $, period $ T = \frac{2\pi}{1} = 2\pi $.
## Step2: Determine desired amplitude and period
We need a function with half the amplitude ($ \frac{2}{2} = 1 $) and half the period ($ \frac{2\pi}{2} = \pi $).
## Step3: Find $ A $ and $ B $ for new function
- Amplitude: $ |A| = 1 $, so $ A = 1 $ (or -1, but we'll consider positive for simplicity).
- Period: $ \frac{2\pi}{|B|} = \pi \implies |B| = 2 $, so $ B = 2 $ (or -2).
## Step4: Identify the function
The function with $ A = 1 $ and $ B = 2 $ is $ y = \sin(2x) $.
# Answer:
B) $ \sin(2x) $

### Question 32
# Explanation:
## Step1: Substitute $ m $ from first equation into second
We have $ m = 3n + 5 $ and $ 2m = 5p + 20 $. Substitute $ m $ into the second equation: $ 2(3n + 5) = 5p + 20 $
## Step2: Simplify the equation
Expand the left side: $ 6n + 10 = 5p + 20 $
## Step3: Solve for $ n $
Subtract 10 from both sides: $ 6n = 5p + 10 $
Divide both sides by 6: $ n = \frac{5p + 10}{6} $
# Answer:
B) $ n = \frac{(5p + 10)}{6} $

### Question 33
# Explanation:
## Step1: Analyze the sine value
Given $ \sin(\theta) = 0.5 $, so $ \theta = 30^\circ $ (since it's a right triangle, angles are between $ 0^\circ $ and $ 90^\circ $ for non - right angles).
## Step2: Recall angles in a right triangle
In a right triangle, the sum of angles is $ 180^\circ $, and one angle is $ 90^\circ $. If one non - right angle is $ 30^\circ $, the other non - right angle is $ 180^\circ-90^\circ - 30^\circ=60^\circ $.
## Step3: Verify with area (optional)
The area of a right triangle is $ \frac{1}{2}ab $ (where $ a $ and $ b $ are the legs). If the angles are $ 30^\circ $, $ 60^\circ $, $ 90^\circ $ and hypotenuse $ c = 7 $, let the legs be $ a=\frac{c}{2}=3.5 $ (opposite $ 30^\circ $) and $ b = \frac{c\sqrt{3}}{2}=\frac{7\sqrt{3}}{2} $. Then the area is $ \frac{1}{2}\times3.5\times\frac{7\sqrt{3}}{2}=\frac{49\sqrt{3}}{8} $, which matches the given area.
# Answer:
A) $ 30^\circ $, $ 60^\circ $, $ 90^\circ $

### Question 34
# Explanation:
## Step1: Set up the system of equations
We have two circle equations:
1. $ x^{2}+y^{2}=2 $
2. $ (x - 2)^{2}+y^{2}=2 $
## Step2: Subtract the first equation from the second
$ (x - 2)^{2}+y^{2}-(x^{2}+y^{2})=2 - 2 $
Expand $ (x - 2)^{2}=x^{2}-4x + 4 $, so:
$ x^{2}-4x + 4+y^{2}-x^{2}-y^{2}=0 $
Simplify to get: $ - 4x+4 = 0\implies x = 1 $
## Step3: Substitute $ x = 1 $ into the first equation
Substitute $ x = 1 $ into $ x^{2}+y^{2}=2 $: $ 1 + y^{2}=2\implies y^{2}=1\implies y=\pm1 $
So the intersection points are $ (1,1) $ and $ (1, - 1) $. Among the options, $ (1,1) $ is a solution.
# Answer:
B) $ (1,1) $

### Question 35
# Explanation:
## Step1: Recall the formula for the space diagonal of a rectangular prism
The formula for the length of the space diagonal $ d $ of a rectangular prism with length $ l $, width $ w $, and height $ h $ is $ d=\sqrt{l^{2}+w^{2}+h^{2}} $
## Step2: Substitute the values
Given $ l = 3 \, \text{m} $, $ w = 4 \, \text{m} $, $ h = 12 \, \text{m} $
$ d=\sqrt{3^{2}+4^{2}+12^{2}}=\sqrt{9 + 16+144}=\sqrt{169}=13 \, \text{m} $
# Answer:
A) $ 13 \, \text{m} $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 30

Step1: Calculate volume of large cube

Step2: Calculate volumes of small cubes

Step3: Calculate total volume of small cubes

Step4: Calculate unoccupied volume

Step1: Recall amplitude and period of sine function

Step2: Determine desired amplitude and period

Step3: Find \( A \) and \( B \) for new function

Step4: Identify the function

Step1: Substitute \( m \) from first equation into second

Step2: Simplify the equation

Step3: Solve for \( n \)

Answer:

Question 31