Question
simplify the polynomial
(7 - 4x^3 + 3x^2) + (5x^2 - 10 + 2x^3)
(7 - 4x^3 + 3x^2) + (5x^2 - 10 + 2x^3)
Answers
Answered by
GPT 3.5
Combining like terms, we have:
(7 - 4x^3 + 3x^2) + (5x^2 - 10 + 2x^3)
= 7 + 5x^2 - 10 + 3x^2 - 4x^3 + 2x^3
= (7 - 10) + (5x^2 + 3x^2) + (- 4x^3 + 2x^3)
= -3 + 8x^2 - 2x^3
So the simplified polynomial is -3 + 8x^2 - 2x^3.
(7 - 4x^3 + 3x^2) + (5x^2 - 10 + 2x^3)
= 7 + 5x^2 - 10 + 3x^2 - 4x^3 + 2x^3
= (7 - 10) + (5x^2 + 3x^2) + (- 4x^3 + 2x^3)
= -3 + 8x^2 - 2x^3
So the simplified polynomial is -3 + 8x^2 - 2x^3.
Answer
what is the surface area of the rectangular prism?
base = 8, width = 5, height = 5
185
170
220
210
base = 8, width = 5, height = 5
185
170
220
210
Answered by
GPT 3.5
The surface area of a rectangular prism can be found by adding up the areas of all six faces.
The formula to calculate the surface area is given by:
2 * (base * width + base * height + width * height)
Substituting the given values, we have:
2 * (8 * 5 + 8 * 5 + 5 * 5) = 2 * (40 + 40 + 25) = 2 * 105 = 210
Therefore, the surface area of the rectangular prism is 210.
The formula to calculate the surface area is given by:
2 * (base * width + base * height + width * height)
Substituting the given values, we have:
2 * (8 * 5 + 8 * 5 + 5 * 5) = 2 * (40 + 40 + 25) = 2 * 105 = 210
Therefore, the surface area of the rectangular prism is 210.
Answer
simplify 4(xy)^4
Answered by
GPT 3.5
To simplify 4(xy)^4, we can expand the expression using the exponent rule (a^m)^n = a^(m*n).
Thus, 4(xy)^4 becomes 4 * x^4 * y^4.
Therefore, the simplified form of 4(xy)^4 is 4x^4y^4.
Thus, 4(xy)^4 becomes 4 * x^4 * y^4.
Therefore, the simplified form of 4(xy)^4 is 4x^4y^4.
Answer
AC is a diameter of circle O, and m<AOB = 56. what is mBC?
Answered by
GPT 3.5
In a circle, the angle formed by a diameter and a chord that intersects the diameter is always 90 degrees. This is a property of circles.
Therefore, in triangle ABC, with AC as the diameter of circle O, angle BAC is a right angle.
Since the sum of the angles in a triangle is 180 degrees, and we know that angle BAC is 90 degrees, we can deduce that the sum of angles B and C is also equal to 90 degrees.
Therefore, mBC = 90 degrees.
Therefore, in triangle ABC, with AC as the diameter of circle O, angle BAC is a right angle.
Since the sum of the angles in a triangle is 180 degrees, and we know that angle BAC is 90 degrees, we can deduce that the sum of angles B and C is also equal to 90 degrees.
Therefore, mBC = 90 degrees.
Answer
simplify (-3a)(-2b)
Answered by
GPT 3.5
To simplify (-3a)(-2b), we can multiply the terms inside the parentheses:
(-3a)(-2b) = (-3)(-2)(ab) = 6ab
Therefore, the simplified expression is 6ab.
(-3a)(-2b) = (-3)(-2)(ab) = 6ab
Therefore, the simplified expression is 6ab.
Answer
a piece of machinery depreciates $7000 the first year, $6800 the second year, and $6600 the third year. if the rate of depreciation is constant, what is the amount of depreciation of the piece of machinery in the sixth year?
Answered by
GPT 3.5
If the rate of depreciation is constant, we can determine the amount of depreciation in the sixth year by subtracting the depreciation of each previous year from the depreciation of the first year.
First, we calculate the difference in depreciation between the first and second year:
$7000 - $6800 = $200
Next, we calculate the difference in depreciation between the second and third year:
$6800 - $6600 = $200
Since the difference in depreciation is constant at $200 per year, we can determine the depreciation in the sixth year by subtracting this amount from the depreciation in the first year:
$7000 - ($200 * 3) = $7000 - $600 = $6400
Therefore, the amount of depreciation of the piece of machinery in the sixth year would be $6400.
First, we calculate the difference in depreciation between the first and second year:
$7000 - $6800 = $200
Next, we calculate the difference in depreciation between the second and third year:
$6800 - $6600 = $200
Since the difference in depreciation is constant at $200 per year, we can determine the depreciation in the sixth year by subtracting this amount from the depreciation in the first year:
$7000 - ($200 * 3) = $7000 - $600 = $6400
Therefore, the amount of depreciation of the piece of machinery in the sixth year would be $6400.
Answer
a piece of machinery depreciates $7000 the first year, $6800 the second year, and $6600 the third year. if the rate of depreciation is constant, what is the amount of depreciation of the piece of machinery in the sixth year?
$41,000
$6000
$62,000
$5800
$41,000
$6000
$62,000
$5800
Answered by
GPT 3.5
To determine the amount of depreciation in the sixth year, we can observe the pattern in the depreciation amounts provided.
In the given scenario, the machinery depreciates by $200 each year ($7000 - $6800 = $200, $6800 - $6600 = $200). This suggests a constant rate of depreciation of $200 per year.
To calculate the amount of depreciation in the sixth year, we can start with the depreciation in the first year and subtract $200 for each subsequent year.
Depreciation in the first year: $7000
Depreciation in the second year: $7000 - $200 = $6800
Depreciation in the third year: $6800 - $200 = $6600
Depreciation in the fourth year: $6600 - $200 = $6400
Depreciation in the fifth year: $6400 - $200 = $6200
Depreciation in the sixth year: $6200 - $200 = $6000
Therefore, the amount of depreciation of the piece of machinery in the sixth year is $6000.
In the given scenario, the machinery depreciates by $200 each year ($7000 - $6800 = $200, $6800 - $6600 = $200). This suggests a constant rate of depreciation of $200 per year.
To calculate the amount of depreciation in the sixth year, we can start with the depreciation in the first year and subtract $200 for each subsequent year.
Depreciation in the first year: $7000
Depreciation in the second year: $7000 - $200 = $6800
Depreciation in the third year: $6800 - $200 = $6600
Depreciation in the fourth year: $6600 - $200 = $6400
Depreciation in the fifth year: $6400 - $200 = $6200
Depreciation in the sixth year: $6200 - $200 = $6000
Therefore, the amount of depreciation of the piece of machinery in the sixth year is $6000.
Answer
what is the component form of the vector? (4,4) (1,2)
(3,2)
(4,6)
(3,4)
(5,6)
(3,2)
(4,6)
(3,4)
(5,6)
Answered by
GPT 3.5
To find the component form of the vector, we subtract the corresponding components:
(4, 4) - (1, 2) = (4 - 1, 4 - 2) = (3, 2)
Therefore, the component form of the vector (4, 4) - (1, 2) is (3, 2).
(4, 4) - (1, 2) = (4 - 1, 4 - 2) = (3, 2)
Therefore, the component form of the vector (4, 4) - (1, 2) is (3, 2).
Answer
<BEC = 22. what is <AEB
22
32
42
158
22
32
42
158
Answered by
GPT 3.5
To find the measure of angle AEB, we know that angles on a straight line add up to 180 degrees. Since angle BEC is given as 22 degrees, we can subtract that from 180 to find angle AEB:
180 - 22 = 158
Therefore, the measure of angle AEB is 158 degrees.
180 - 22 = 158
Therefore, the measure of angle AEB is 158 degrees.
Answer
which similarity postulate of theorem, if any, can be used to prove the two triangles similar?
first triangle (b = 26, other sides 16 and 20)
second triangle (base= 13, and the other sides 10 and 8)
SSS Similarity theorem
AA similarity postulate
SAS theorem
the triangles are not similar
first triangle (b = 26, other sides 16 and 20)
second triangle (base= 13, and the other sides 10 and 8)
SSS Similarity theorem
AA similarity postulate
SAS theorem
the triangles are not similar
Answered by
GPT 3.5
To determine if the two triangles are similar and which similarity postulate or theorem can be used to prove their similarity, we need to compare their corresponding sides and angles.
In the given triangles, we have the following side lengths:
First triangle: 16, 20, 26
Second triangle: 8, 10, 13
Since the side lengths are not proportional, we cannot use the SSS Similarity theorem to prove their similarity.
Next, let's compare the corresponding angles:
In the first triangle, we have angle A, angle B, and angle C.
In the second triangle, we have angle X, angle Y, and angle Z.
However, we do not have any information about the angles, so we cannot apply the AA similarity postulate or the SAS theorem.
Therefore, based solely on the information given, we cannot prove that the triangles are similar.
In the given triangles, we have the following side lengths:
First triangle: 16, 20, 26
Second triangle: 8, 10, 13
Since the side lengths are not proportional, we cannot use the SSS Similarity theorem to prove their similarity.
Next, let's compare the corresponding angles:
In the first triangle, we have angle A, angle B, and angle C.
In the second triangle, we have angle X, angle Y, and angle Z.
However, we do not have any information about the angles, so we cannot apply the AA similarity postulate or the SAS theorem.
Therefore, based solely on the information given, we cannot prove that the triangles are similar.
Answer
What is the length of BD if CB = 7x + 12, BD = 6x + 14, and CD = 65?
7
3
32
33
7
3
32
33
Answered by
GPT 3.5
To find the length of BD, we need to set up an equation using the lengths of CB, BD, and CD.
CD is given as 65.
CB = 7x + 12
BD = 6x + 14
We can set up the equation: CB + BD = CD
(7x + 12) + (6x + 14) = 65
Combining like terms, we have: 13x + 26 = 65
Subtracting 26 from both sides, we get: 13x = 39
Dividing both sides by 13, we find: x = 3
Now that we know x = 3, we can substitute it back into BD = 6x + 14:
BD = 6(3) + 14
BD = 18 + 14
BD = 32
Therefore, the length of BD is 32.
CD is given as 65.
CB = 7x + 12
BD = 6x + 14
We can set up the equation: CB + BD = CD
(7x + 12) + (6x + 14) = 65
Combining like terms, we have: 13x + 26 = 65
Subtracting 26 from both sides, we get: 13x = 39
Dividing both sides by 13, we find: x = 3
Now that we know x = 3, we can substitute it back into BD = 6x + 14:
BD = 6(3) + 14
BD = 18 + 14
BD = 32
Therefore, the length of BD is 32.
Answer
Solve |x| + 5 > 8
Answered by
GPT 3.5
We will solve the inequality |x| + 5 > 8 by considering two cases:
Case 1: x is positive or zero (x >= 0)
If x >= 0, then |x| = x. Substituting into the inequality:
x + 5 > 8
Subtracting 5 from both sides:
x > 3
Therefore, in this case, the solution is x > 3.
Case 2: x is negative (x < 0)
If x < 0, then |x| = -x. Substituting into the inequality:
-x + 5 > 8
Subtracting 5 from both sides:
-x > 3
Dividing both sides by -1, we need to reverse the inequality symbol:
x < -3
Therefore, in this case, the solution is x < -3.
Combining the solutions from both cases, we have:
x > 3 or x < -3.
The solution set for |x| + 5 > 8 is x > 3 or x < -3.
Case 1: x is positive or zero (x >= 0)
If x >= 0, then |x| = x. Substituting into the inequality:
x + 5 > 8
Subtracting 5 from both sides:
x > 3
Therefore, in this case, the solution is x > 3.
Case 2: x is negative (x < 0)
If x < 0, then |x| = -x. Substituting into the inequality:
-x + 5 > 8
Subtracting 5 from both sides:
-x > 3
Dividing both sides by -1, we need to reverse the inequality symbol:
x < -3
Therefore, in this case, the solution is x < -3.
Combining the solutions from both cases, we have:
x > 3 or x < -3.
The solution set for |x| + 5 > 8 is x > 3 or x < -3.
Answer
(6x - 3)
Answered by
GPT 3.5
The expression (6x - 3) cannot be simplified any further.
Answer
On Tuesday alex earned $45.50 this is $17.25 less than he earned monday. which equation can be used to find m, the amount alex earned monday?
45.50 - m = 17.25
17.25 - m = 45.50
m + 45.50 = 17.25
m - 17.25 = 45.50
45.50 - m = 17.25
17.25 - m = 45.50
m + 45.50 = 17.25
m - 17.25 = 45.50
Answered by
GPT 3.5
The correct equation that can be used to find the amount Alex earned on Monday is:
m - 17.25 = 45.50
Since he earned $17.25 less on Tuesday than on Monday, we need to subtract $17.25 from the amount he earned on Monday (m) to get the amount he earned on Tuesday ($45.50).
m - 17.25 = 45.50
Since he earned $17.25 less on Tuesday than on Monday, we need to subtract $17.25 from the amount he earned on Monday (m) to get the amount he earned on Tuesday ($45.50).
Answer
Solve -5(8x + 5) + 3x = 2
Answered by
GPT 3.5
Let's simplify and solve the equation:
-5(8x + 5) + 3x = 2
First, distribute -5 to the terms within the parentheses:
-40x - 25 + 3x = 2
Next, combine like terms:
-40x + 3x - 25 = 2
Simplifying further:
-37x - 25 = 2
Now, isolate the variable by adding 25 to both sides:
-37x = 27
Finally, divide both sides by -37 to solve for x:
x = 27 / -37
The solution is:
x = -27/37
-5(8x + 5) + 3x = 2
First, distribute -5 to the terms within the parentheses:
-40x - 25 + 3x = 2
Next, combine like terms:
-40x + 3x - 25 = 2
Simplifying further:
-37x - 25 = 2
Now, isolate the variable by adding 25 to both sides:
-37x = 27
Finally, divide both sides by -37 to solve for x:
x = 27 / -37
The solution is:
x = -27/37
Answer
alex and her brother are painting a sandbox shaped like a rectangular prism. they want to paint only the outsides of the four rectangular sides. how many square feet will be painted?
base = 6, width = 5, height = 1
24ft
22ft
20ft
41ft
base = 6, width = 5, height = 1
24ft
22ft
20ft
41ft
Answered by
GPT 3.5
To find the area of the four rectangular sides that will be painted, we need to calculate the perimeter of the base and multiply it by the height.
Perimeter of the base = 2(base + width) = 2(6 + 5) = 2(11) = 22 feet
Area of the four rectangular sides = Perimeter of the base * height = 22 * 1 = 22 square feet
Therefore, the area that will be painted is 22 square feet.
Perimeter of the base = 2(base + width) = 2(6 + 5) = 2(11) = 22 feet
Area of the four rectangular sides = Perimeter of the base * height = 22 * 1 = 22 square feet
Therefore, the area that will be painted is 22 square feet.
Answer
a rubber ball makes a sequence of bounces. on its first bounce, the ball reached a height of 10 feet. each bounce is 4/5 as high as the preceding bounce. to the nearest tenth of a foot, what is the height of the eighth bounce?
8.0ft
9.8ft
2.1ft
6.4ft
8.0ft
9.8ft
2.1ft
6.4ft
Answered by
GPT 3.5
To find the height of the eighth bounce, we can use the information given: each bounce is 4/5 as high as the preceding bounce.
Starting with the height of the first bounce, which is 10 feet, we can repeatedly multiply each bounce's height by 4/5 to find the height of the next bounce.
First bounce: 10 feet
Second bounce: (4/5) * 10 = 8 feet
Third bounce: (4/5) * 8 = 6.4 feet
Fourth bounce: (4/5) * 6.4 = 5.12 feet
Fifth bounce: (4/5) * 5.12 = 4.096 feet
Sixth bounce: (4/5) * 4.096 = 3.2768 feet
Seventh bounce: (4/5) * 3.2768 = 2.62144 feet
Eighth bounce: (4/5) * 2.62144 ≈ 2.09715 feet
Rounding to the nearest tenth of a foot, the height of the eighth bounce is approximately 2.1 feet.
Therefore, the nearest answer choice is 2.1ft.
Starting with the height of the first bounce, which is 10 feet, we can repeatedly multiply each bounce's height by 4/5 to find the height of the next bounce.
First bounce: 10 feet
Second bounce: (4/5) * 10 = 8 feet
Third bounce: (4/5) * 8 = 6.4 feet
Fourth bounce: (4/5) * 6.4 = 5.12 feet
Fifth bounce: (4/5) * 5.12 = 4.096 feet
Sixth bounce: (4/5) * 4.096 = 3.2768 feet
Seventh bounce: (4/5) * 3.2768 = 2.62144 feet
Eighth bounce: (4/5) * 2.62144 ≈ 2.09715 feet
Rounding to the nearest tenth of a foot, the height of the eighth bounce is approximately 2.1 feet.
Therefore, the nearest answer choice is 2.1ft.
Answer
which ordered pair is a solution of this equation? 9-x/7 = y
(2,1)
(-2,0)
(9,81)
(7,-7)
(2,1)
(-2,0)
(9,81)
(7,-7)
Answered by
GPT 3.5
To determine which ordered pair is a solution of the equation 9 - x/7 = y, we substitute the x- and y-values into the equation and check if it holds true.
Let's check each option:
Option 1: (2, 1)
Substituting x = 2 and y = 1 into the equation:
9 - 2/7 = 1
8.7142857 ≠ 1
Option 2: (-2, 0)
Substituting x = -2 and y = 0 into the equation:
9 - (-2)/7 = 0
9.2857143 ≠ 0
Option 3: (9, 81)
Substituting x = 9 and y = 81 into the equation:
9 - 9/7 = 81
8.2857143 ≠ 81
Option 4: (7, -7)
Substituting x = 7 and y = -7 into the equation:
9 - 7/7 = -7
8.0000000 = -7
From the options given, only the ordered pair (7, -7) satisfies the equation.
Therefore, the solution is (7, -7).
Let's check each option:
Option 1: (2, 1)
Substituting x = 2 and y = 1 into the equation:
9 - 2/7 = 1
8.7142857 ≠ 1
Option 2: (-2, 0)
Substituting x = -2 and y = 0 into the equation:
9 - (-2)/7 = 0
9.2857143 ≠ 0
Option 3: (9, 81)
Substituting x = 9 and y = 81 into the equation:
9 - 9/7 = 81
8.2857143 ≠ 81
Option 4: (7, -7)
Substituting x = 7 and y = -7 into the equation:
9 - 7/7 = -7
8.0000000 = -7
From the options given, only the ordered pair (7, -7) satisfies the equation.
Therefore, the solution is (7, -7).
Answer
solve (a - 3)(a - 4) = 0
Answered by
GPT 3.5
To solve the equation (a - 3)(a - 4) = 0, we can set each factor equal to zero and solve for a.
(a - 3) = 0 or (a - 4) = 0
If (a - 3) = 0, then we add 3 to both sides:
a - 3 + 3 = 0 + 3
a = 3
If (a - 4) = 0, then we add 4 to both sides:
a - 4 + 4 = 0 + 4
a = 4
Therefore, the solutions to the equation (a - 3)(a - 4) = 0 are a = 3 and a = 4.
(a - 3) = 0 or (a - 4) = 0
If (a - 3) = 0, then we add 3 to both sides:
a - 3 + 3 = 0 + 3
a = 3
If (a - 4) = 0, then we add 4 to both sides:
a - 4 + 4 = 0 + 4
a = 4
Therefore, the solutions to the equation (a - 3)(a - 4) = 0 are a = 3 and a = 4.
Answer
find the measure of one of the interior angles of a regular polygon with 8 sides.
sun of the measures of any interior angles of any n-gon = (n-2)180
measure of an interior angle = (n-2)180/n
measure of an exterior angle = 360/n
a = 1/2ap
1080 degrees
157.5 degrees
135 degrees
202.5 degrees
sun of the measures of any interior angles of any n-gon = (n-2)180
measure of an interior angle = (n-2)180/n
measure of an exterior angle = 360/n
a = 1/2ap
1080 degrees
157.5 degrees
135 degrees
202.5 degrees
Answered by
GPT 3.5
To find the measure of one of the interior angles of a regular polygon with 8 sides, we can use the formula:
Measure of an interior angle = (n-2)180/n
Plugging in the value of n (number of sides):
Measure of an interior angle = (8-2)180/8
Measure of an interior angle = 6*180/8
Measure of an interior angle = 135 degrees
Therefore, the measure of one of the interior angles of a regular polygon with 8 sides is 135 degrees.
Measure of an interior angle = (n-2)180/n
Plugging in the value of n (number of sides):
Measure of an interior angle = (8-2)180/8
Measure of an interior angle = 6*180/8
Measure of an interior angle = 135 degrees
Therefore, the measure of one of the interior angles of a regular polygon with 8 sides is 135 degrees.
Answer
nathan is making chocolates. he starts by melting a chocolate shaped like a rectangular prism. then he pours the melted chocolate into molds shaped like rectangular prisms. how many chocolates can nathan make with a block of chocolate?
chocolate mold measurements: base = 3, width = 1, height = 2
block of chocolate measurements: base = 20, width = 3cm height = 22cm
answers
440
220
22
110
chocolate mold measurements: base = 3, width = 1, height = 2
block of chocolate measurements: base = 20, width = 3cm height = 22cm
answers
440
220
22
110
Answered by
GPT 3.5
To determine how many chocolates Nathan can make with the block of chocolate, we need to compare the volume of the block of chocolate to the volume of one chocolate mold.
The volume of a rectangular prism can be calculated by multiplying the length, width, and height.
Volume of chocolate mold = base * width * height = 3 * 1 * 2 = 6 cubic cm
Volume of block of chocolate = base * width * height = 20 * 3 * 22 = 1320 cubic cm
To find the number of chocolates Nathan can make, we divide the volume of the block of chocolate by the volume of one chocolate mold:
Number of chocolates = Volume of block of chocolate / Volume of chocolate mold = 1320 / 6 = 220
Therefore, Nathan can make 220 chocolates with the block of chocolate. Thus, the answer is 220.
The volume of a rectangular prism can be calculated by multiplying the length, width, and height.
Volume of chocolate mold = base * width * height = 3 * 1 * 2 = 6 cubic cm
Volume of block of chocolate = base * width * height = 20 * 3 * 22 = 1320 cubic cm
To find the number of chocolates Nathan can make, we divide the volume of the block of chocolate by the volume of one chocolate mold:
Number of chocolates = Volume of block of chocolate / Volume of chocolate mold = 1320 / 6 = 220
Therefore, Nathan can make 220 chocolates with the block of chocolate. Thus, the answer is 220.
Answer
an office spends $350 per month on office supplies. the manager wants the company to decease office supply costs by 16%. how much money would that save each month?
$334
$54
$294
$56
$334
$54
$294
$56
Answered by
GPT 3.5
To determine the amount of money saved each month by decreasing office supply costs by 16%, we need to calculate 16% of $350.
16% of $350 = 0.16 * $350 = $56
Therefore, decreasing office supply costs by 16% would save $56 each month. Thus, the correct answer is $56.
16% of $350 = 0.16 * $350 = $56
Therefore, decreasing office supply costs by 16% would save $56 each month. Thus, the correct answer is $56.
Answer
determine the number of solutions for the system of linear equations
y = -10x + 5
300 = -15x + 60y
y = -10x + 5
300 = -15x + 60y