The sentences, when converted into a sentence having the form "If P , then Q " are:
If the determinant of a matrix is not zero, then the matrix is invertible.If a function is integrable, then it is continuous.If an integer is divisible by 8, then it is divisible by 4.If a series converges absolutely, then it converges.If a function is continuous, then it is integrable.If people agree with me, then I feel I must be wrong.How to convert the sentences ?To transform these sentences into the "If P, then Q" format, we will identify the condition (P) and the result or consequence (Q) in each sentence.
A matrix is invertible provided that its determinant is not zero."
The condition here is "its determinant is not zero", and the result is "the matrix is invertible". Thus, we can rephrase the sentence as: "If the determinant of a matrix is not zero, then the matrix is invertible."
"For a function to be integrable, it is necessary that it is continuous."
Here, the condition is that "the function is integrable", and the result is "it is continuous". So, we can rephrase the sentence as: "If a function is integrable, then it is continuous."
"An integer is divisible by 8 only if it is divisible by 4."
In this sentence, "an integer is divisible by 8" is the condition, and "it is divisible by 4" is the result. We then say, "If an integer is divisible by 8, then it is divisible by 4."
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If the determinant of a matrix is not zero, then the matrix is invertible.
If a function is continuous, then it is necessary for it to be integrable.If an integer is divisible by 4, then it is divisible by 8.
If a series converges absolutely, then the series converges. If a function is continuous, then it is integrable.If people agree with me, then I feel I must be wrong.
A complete sentence has a subject and predicate and should contain at least one independent clause.
An independent clause is a clause that can stand on its own as a complete sentence.
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2.1 Convert the following: 1. 10g to Kg. 2. 32km to meter. 3. 12 m² to mm²
4. 50000mm³ to m³
5. 2,36hrs to hrs, minutes and seconds
2.2 The distance between town A and town B is 16500m. What is the distance exactly halfway between the towns in Km?
10g is equal to 0.01 Kg.
32km is equal to 32,000 meters.
12 m² is equal to 12,000 mm².
50,000mm³ is equal to 0.05 m³.
2.36hrs is equal to 2 hours, 21 minutes, and 36 seconds.
The distance exactly halfway between town A and town B is 8.25 km.
To convert grams to kilograms, divide the given value by 1000 since there are 1000 grams in a kilogram.
To convert kilometers to meters, multiply the given value by 1000 since there are 1000 meters in a kilometer.
To convert square meters to square millimeters, multiply the given value by 1,000,000 since there are 1,000,000 square millimeters in a square meter.
To convert cubic millimeters to cubic meters, divide the given value by 1,000,000,000 since there are 1,000,000,000 cubic millimeters in a cubic meter.
To convert hours to hours, minutes, and seconds, the given value can be expressed as 2 hours and 0.36 hours. The decimal part represents the minutes and seconds. Multiply 0.36 by 60 to get 21.6 minutes, and then convert 0.6 minutes to seconds, which is 36 seconds.
For the second part of the question, to find the distance exactly halfway between town A and town B, divide the total distance (16500m) by 2 to get 8250m. Since the answer should be in kilometers, divide 8250 by 1000 to get 8.25 Km.
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Solve each equation.
log₁₀ 0.001=x
The equation log₁₀ 0.001 = x can be solved by rewriting it in exponential form: 10^x = 0.001. Taking the logarithm of both sides with base 10, we find that x = -3.
To solve the equation log₁₀ 0.001 = x, we need to convert it to exponential form. The logarithm with base 10 is equivalent to an exponentiation with base 10. In this case, the logarithm of 0.001 with base 10 is equal to x.
To rewrite the equation in exponential form, we raise 10 to the power of both sides: 10^x = 0.001. This equation states that 10 raised to the power of x is equal to 0.001.
To find the value of x, we need to determine the exponent that yields 0.001 when 10 is raised to that power. By calculating the value of 10^x, we find that x = -3.
Therefore, the solution to the equation log₁₀ 0.001 = x is x = -3. This means that the logarithm of 0.001 with base 10 is equal to -3.
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Percentage. Mixing 54 kg of salt water with 76 kg clear water is created water containing 2.7% salt. How many percent salt water contains salt?
A solution is made by combining 54 kg of salt water with 76 kg of clear water, producing water that contains 2.7 percent salt. The percentage of salt in the saltwater is 41.5%.
The problem is asking us to calculate the percentage of salt present in saltwater. We are given the amount of saltwater and clear water used to create a solution with 2.7% salt. 54 kg of salt water and 76 kg of clear water are combined to make a solution. We want to know what percentage of the salt water is salt.
As we know, the percentage of salt in the saltwater is (mass of salt / total mass of saltwater) × 100. Let us assume that the mass of salt present in the salt water is x kg. Therefore, the mass of salt water (salt + water) is 54 kg. So, the mass of salt is x kg and the mass of water is (54 - x) kg. Since the solution contains 2.7% salt, we can write:
(mass of salt / total mass of saltwater) × 100 = 2.7%. Also, we have the total mass of the solution:
The total mass of solution = Mass of salt water + mass of clear water = 54 + 76 = 130 kg.
Now we can write the equation as: [tex]\frac{x}{54} \times 100 = 2.7 \%[/tex]. And we know that the total mass of the solution is 130 kg:
x + (54 - x) = 130 kg. By solving the above equation we get,x = 30.6 kg. So, the percentage of salt in the saltwater is [tex]\frac{30.6 }{54} \times 100 = 56.67 \%[/tex]. Approximately 56.67% of the saltwater is salt.
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The figure shows two kayakers pulling a raft. One kayaker pulls with force vector F sub 1 equals open angled bracket 190 comma 160 close angled bracket comma and the other kayaker pulls with force vector F sub 2 equals open angled bracket 128 comma negative 121 close angled bracket period
two vectors F sub 1 and F sub 2 that share an initial point located on a raft, F sub 1 points right and up where its terminal point is at a kayak, F sub 2 points left and down where its terminal point is at another kayak
What is the angle between the kayakers? Round your answer to the nearest degree. (2 points)
78°
83°
86°
80°
Answer: Therefore, the angle between the kayakers is approximately 63 degrees. The closest answer choice is 78°.
Step-by-step explanation:
To find the angle between the kayakers, we can use the dot product formula:
F sub 1 · F sub 2 = ||F sub 1|| ||F sub 2|| cos θ
where · denotes the dot product, || || denotes the magnitude, and θ is the angle between the two vectors.
First, we need to find the magnitudes of F sub 1 and F sub 2:
||F sub 1|| = sqrt(190^2 + 160^2) = 247.79
||F sub 2|| = sqrt(128^2 + (-121)^2) = 170.10
Next, we need to find the dot product of F sub 1 and F sub 2:
F sub 1 · F sub 2 = (190)(128) + (160)(-121) = -12080
Substituting these values into the dot product formula, we get:
-12080 = (247.79)(170.10) cos θ
Solving for cos θ, we get:
cos θ = -0.424
Taking the inverse cosine of both sides, we get:
θ ≈ 116.8°
However, this is the angle between the two vectors in standard position (i.e., with initial points at the origin). To find the angle between the kayakers, we need to subtract this angle from 180°:
180° - θ ≈ 63.2°
A 3500 lbs car rests on a hill inclined at 6◦ from the horizontal. Find the magnitude
of the force required (ignoring friction) to prevent the car from rolling down the hill. (Round
your answer to 2 decimal places)
The magnitude of the force required to prevent the car from rolling down the hill is 1578.88 Newton.
How to calculate the magnitude of the force?In accordance with Newton's Second Law of Motion, the force acting on this car is equal to the horizontal component of the force (Fx) that is parallel to the slope:
Fx = mgcosθ
Fx = Fcosθ
Where:
F represents the force.m represents the mass of a physical object.g represents the acceleration due to gravity.Note: 3500 lbs to kg = 3500/2.205 = 1587.573 kg
By substituting the given parameters into the formula for the horizontal component of the force (Fx), we have;
Fx = 1587.573cos(6)
Fx = 1578.88 Newton.
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The magnitude of the force required to prevent the car from rolling down the hill is approximately 367.01 lbs.
To find the magnitude of the force required to prevent the car from rolling down the inclined hill, we can analyze the forces acting on the car.
The weight of the car acts vertically downward with a magnitude of 3500 lbs. We can decompose this weight into two components: one perpendicular to the incline and one parallel to the incline.
The component perpendicular to the incline can be calculated as W_perpendicular = 3500 * cos(6°).
The component parallel to the incline represents the force that tends to make the car roll down the hill. To prevent this, an equal and opposite force is required, which is the force we need to find.
Since we are ignoring friction, the force required to prevent rolling is equal to the parallel component of the weight: F_required = 3500 * sin(6°).
Calculating this value gives:
F_required = 3500 * sin(6°) ≈ 367.01 lbs (rounded to 2 decimal places).
Therefore, the magnitude of the force required to prevent the car from rolling down the hill is approximately 367.01 lbs.
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Find the solution of the given initial value problem. y (4)
−12y ′′′
+36y ′′
=0
y(1)=14+e 6
,y ′
(1)=9+6e 6
,y ′′
(1)=36e 6
,y ′′′
(1)=216e 6
.
y(t)=∫
How does the solution behave as t→[infinity] ?
The highest degree of the equation is 3. As t approaches infinity, the value of the equation also tends to infinity as the degree of the equation is odd.
The given initial value problem is:
y(4) − 12y′′′ + 36y′′ = 0,
y(1) = 14 + e6,
y′(1) = 9 + 6e6,
y′′(1) = 36e6,
y′′′(1) = 216e6
To find the solution of the given initial value problem, we proceed as follows:
Let y(t) = et
Now, y′(t) = et,
y′′(t) = et,
y′′′(t) = et and
y(4)(t) = et
Substituting the above values in the given equation, we have:
et − 12et + 36et = 0et(1 − 12 + 36)
= 0et
= 0 and
y(t) = c1 + c2t + c3t² + c4t³
Where c1, c2, c3, and c4 are constants.
To determine these constants, we apply the given initial conditions.
y(1) = 14 + e6 gives
c1 + c2 + c3 + c4 = 14 + e6y′(1)
= 9 + 6e6 gives c2 + 2c3 + 3c4 = 9 + 6e6y′′(1)
= 36e6 gives 2c3 + 6c4 = 36e6
y′′′(1) = 216e6
gives 6c4 = 216e6
Solving these equations, we get:
c1 = 14, c2 = 12 + 5e6,
c3 = 12e6,
c4 = 36e6
Thus, the solution of the given initial value problem is:
y(t) = 14 + (12 + 5e6)t + 12e6t² + 36e6t³y(t)
= 36t³ + 12e6t² + (12 + 5e6)t + 14
Hence, the solution of the given initial value problem is 36t³ + 12e6t² + (12 + 5e6)t + 14.
As t approaches infinity, the behavior of the solution can be determined by analyzing the highest degree of the equation.
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The related function is decreasing when x<0 and the zeros are -2 and 2
Answer:
Step-by-step explanation:
If the related function is decreasing when x < 0, it means that as x decreases (moves to the left on the x-axis), the corresponding y-values of the function decrease as well. In other words, the function is getting smaller as x becomes more negative.
Given that the zeros of the function are -2 and 2, it means that when x = -2 or x = 2, the function evaluates to zero. This means that the graph of the function intersects the x-axis at x = -2 and x = 2.
Based on this information, we can conclude that the related function starts from positive values, decreases as x moves to the left (x < 0), and intersects the x-axis at x = -2 and x = 2.
Use the properties of logarithms to simplify and solve each equation. Round to the nearest thousandth.
3 ln x-ln 2=4
The solution to the equation 3 ln x - ln 2 = 4 is x ≈ 4.937.
To solve the equation 3 ln x - ln 2 = 4, we can use the properties of logarithms.
First, we can combine the two logarithms on the left side using the quotient property of logarithms. According to this property, ln(a) - ln(b) is equal to ln(a/b):
So, we can rewrite the equation as ln(x^3/2) = 4.
Next, we can convert the logarithmic equation into an exponential equation. The exponential form of ln(x) = y is e^y = x, where, e is the base of the natural logarithm.
Applying this to our equation, we get e^4 = x^3/2.
To isolate x, we can multiply both sides of the equation by 2 and then take the square root of both sides.
2 * e^4 = x^3
x = (2 * e^4)^(1/3)
Rounding to the nearest thousandth, x ≈ 4.937.
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