Final answer:
The division and multiplication of signed numbers follow the same sign rules: two positive numbers yield a positive result, two negatives give a positive, and a pair of numbers with different signs results in a negative outcome.
Explanation:
The rules for the division of signed numbers are similar to the rules for multiplication of signed numbers. Both operations follow the same pattern concerning the signs of the numbers involved:
When two positive numbers are involved, the result is positive (e.g., [tex]\frac{2}{1} = 2[/tex] or 2 x 1 = 2).
When two negative numbers are involved, the result is also positive (e.g., [tex]\frac{-2}{-1} = 2[/tex] or (-4) x (-3) = 12).
When one positive and one negative number are involved, irrespective of the operation, the result is negative (e.g., [tex]\frac{-3}{1} = -3[/tex] or (-3) x 2 = -6).
Therefore, both division and multiplication of signed numbers primarily depend on the signs of the numbers to determine the sign of the answer.
help me solve this, Pre-Calculus Question
I only need the last part
Triangle ABC has two known angles. Angle A measures 55 degrees. Angle b measures 30 degrees.What is the measure of angle C?
The GCF of any two even numbers is always even. true or false?
What is the meaning of the term theorem
If f(x) = 3x + 6, which of the following is the inverse of f(x)?
A. f –1(x) = 3x – 6
B. f –1(x) =
C. f –1(x) =
D. f –1(x) = 6 – 3x
To determine the inverse of the function f(x) = 3x + 6, you need to switch 'x' and 'f(x)', isolate f-1(x) on one side of the equation, then solve for f-1(x). The resulting inverse function is f-1(x) = (x - 6)/3.
Explanation:The function given is f(x) = 3x + 6. To find the inverse of this function, we first need to switch 'x' and 'f(x)', giving us: x = 3f-1(x) + 6. Next, we want to isolate f-1(x) on one side of the equation. To do this, we subtract 6 from both sides of the equation resulting in: x - 6 = 3f-1(x). Finally, we divide all terms by 3 to solve for f-1(x), which gives us f-1(x) = (x - 6)/3. So, the correct answer from the options given is not listed.
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What percent of 9.2 is 43.7
The requried 475% of 9.2 is 43.7, as of the given percentage situation.
What is the percentage?The percentage is the ratio of the composition of matter to the overall composition of matter multiplied by 100.
Here,
In the question, we are asked to determine the 43.7 is what percent of 9.2.
So, let the percent be x,
x % of 9.2 = 43.7
x/100 × 9.2 = 43.7
x = 43.7/9.2 × 100%
x = 475%
Thus, the requried 475% of 9.2 is 43.7, as of the given percentage situation.
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the weight of an object on the moon is about .167 of its weight on Earth. How much does a 180 lb astronaut weigh on the moon?
Two small fires are spotted by a ranger from a fire tower 60 feet above ground. The angles of depressions re 11.6° and 9.4°. How far apart are the fires? (The fires are in the same general direction from the tower.)
The driver of a car traveling at 54ft/sec suddenly applies the brakes. The position of the car is s=54t-3t^2, t seconds afyer the driver applies the brakes.
How many seconds after the driver applies the brakes does the car come to a stop
After time [tex]t = 9[/tex] seconds the driver applies the brakes does the car come to a stop.
What is time?" Time is defined as the measurable slot of period in which required action is done."
Formula used
[tex]y = x^{n} \\\\\implies \frac{dy}{dx} = nx^{n-1}[/tex]
According to the question,
Position of the car [tex]'s' = 54t - 3t^{2}[/tex]
[tex]'s'[/tex] represents the distance
[tex]'t'[/tex] is the time in seconds
When driver applies break [tex]v= 0[/tex],
[tex]v = \frac{ds}{dt}[/tex]
Calculate the first derivative with respect to time we get,
[tex]'s' = 54t - 3t^{2}\\\\\implies \frac{ds}{dt} = 54- 6t[/tex]
As [tex]\frac{ds}{dt} =0[/tex] we get the required time as per given condition,
[tex]54-6t =0\\\\\implies 6t =54\\\\\implies t = 9[/tex]
Hence, after time [tex]t = 9[/tex] seconds the driver applies the brakes does the car come to a stop.
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Suppose 8 out of every 20 students are absent from school less than 5 days a year.Predict how many students would be absent less than 5 days a year out of 40,000 students.
Please need help as quick as you can . What is the slope of the line that passes through the points (16, –1) and (–4, 10)? A. -20/11 B. -11/20 C. 11/20 D. 20/11
I keep getting 13. It's supposed to be 3. Can you show me how it's done? 3(a-5) = -6
a construction crew Must build 4 miles of road in one week on Monday they build 1/2 mile of road in one week on Tuesday they build 1/3 mile of road how many more miles of road are left to build
Find the area of the specified region shared by the circles r = 4 and r = 8sin
Mr. Carandang sold a total of 1,790 prints of one of his drawings. Out of all 1,273 unframed prints that he sold, 152 were small and 544 were medium-sized. Out of all of the framed prints that he sold, 23 were small and 42 were extra large. Of the large prints that he sold, 188 were framed and 496 were unframed. Small Medium Large Extra Large Total Framed 23 264 188 42 ? Unframed 152 544 496 81 1,273 Total 175 808 684 123 1,790 Which number is missing from the two-way table?'
The missing number in the two-way table is the total number of framed prints, which is 517. This is calculated by subtracting the total unframed prints from the total prints sold and then summing the known framed prints.
To find the missing number of prints in the table, we need to determine the total number of framed prints sold by Mr. Carandang. We know the following from the table:
Total prints sold: 1,790Unframed prints: 1,273To find the total number of framed prints:
Total framed prints = Total prints - Total unframed prints
Total framed prints = 1,790 - 1,273 = 517
Now we need to sum the known framed prints:
Framed small: 23Framed medium: 264Framed large: 188Framed extra large: 42The total of known framed prints is:
23 + 264 + 188 + 42 = 517
Thus, the total framed prints value was missing and it is 517.
F(x, y) = x2 + y2 + 4x − 4y, x2 + y2 ≤ 81 find the extreme values of f on the region described by the inequality.
The extreme values of a function are the minimum and the maximum values of the function.
The extreme values are: -8 and 131.91
The given parameters are:
[tex]\mathbf{F(x,y) = x^2 + y^2 + 4x - 4y}[/tex]
[tex]\mathbf{x^2 + y^2 \le 81}[/tex]
Find the gradient of F(x,y)
[tex]\mathbf{f_x(x) = 2x + 4}[/tex]
[tex]\mathbf{f_y(y) = 2y - 4}[/tex]
Set to 0, to solve for x and y
[tex]\mathbf{2x + 4 = 0}[/tex]
[tex]\mathbf{2x = -4}[/tex]
[tex]\mathbf{x = -2}[/tex]
[tex]\mathbf{2y - 4 = 0}[/tex]
[tex]\mathbf{2y = 4}[/tex]
[tex]\mathbf{y = 2}[/tex]
So, the critical point is:
[tex]\mathbf{(x,y) = (-2,2)}[/tex]
Also, we have: [tex]\mathbf{x^2 + y^2 \le 81}[/tex]
Calculate the gradients
[tex]\mathbf{g_x = 2x}[/tex]
[tex]\mathbf{g_y = 2y}[/tex]
Equate the gradients as follows:
[tex]\mathbf{f_x = \lambda \cdot g_x}[/tex]
[tex]\mathbf{f_y = \lambda \cdot g_y}[/tex]
So, we have:
[tex]\mathbf{2x + 4 = \lambda \cdot 2x}[/tex]
[tex]\mathbf{2y - 4 = \lambda \cdot 2y}[/tex]
Make [tex]\mathbf{\lambda}[/tex] the subject, in the above equations
[tex]\mathbf{\lambda = \frac{x + 2}{x}}[/tex]
[tex]\mathbf{\lambda = \frac{y - 2}{y}}[/tex]
Equate the above equations, so we have:
[tex]\mathbf{\frac{x + 2}{x} = \frac{y - 2}{y} }[/tex]
Cross multiply
[tex]\mathbf{xy + 2y = xy - 2x}[/tex]
Subtract xy from both sides
[tex]\mathbf{2y =- 2x}[/tex]
Divide both sides by 2
[tex]\mathbf{y =- x}[/tex]
Substitute [tex]\mathbf{y =- x}[/tex] in [tex]\mathbf{x^2 + y^2 = 81}[/tex]
[tex]\mathbf{x^2 + (-x)^2 = 81}[/tex]
[tex]\mathbf{x^2 + x^2 = 81}[/tex]
[tex]\mathbf{2x^2 = 81}[/tex]
Divide both sides by 2
[tex]\mathbf{x^2 = \frac{81}{2}}[/tex]
Take square roots
[tex]\mathbf{x = \±\frac{9}{\sqrt2}}[/tex]
Rationalize
[tex]\mathbf{x = \±\frac{9\sqrt2}{2}}[/tex]
Recall that: [tex]\mathbf{y =- x}[/tex]
So, we have:
[tex]\mathbf{y = \±\frac{9\sqrt2}{2}}[/tex]
The ordered pairs are:
[tex]\mathbf{(x,y) = \{(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}),(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2})\}}[/tex]
Hence, the points are:
[tex]\mathbf{(x,y) = \{(-2,2),(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}),(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2})\}}[/tex]
Substitute these values in [tex]\mathbf{F(x,y) = x^2 + y^2 + 4x - 4y}[/tex]
[tex]\mathbf{F(2,2) = (-2)^2 + 2^2 + 4(-2) - 4(2) =-8}[/tex]
[tex]\mathbf{F(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}) = (\frac{9\sqrt2}{2})^2 + (-\frac{9\sqrt2}{2})^2 + 4(\frac{9\sqrt2}{2}) - 4(-\frac{9\sqrt2}{2}) =131.91}[/tex]
[tex]\mathbf{F(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2}) = (-\frac{9\sqrt2}{2})^2 + (\frac{9\sqrt2}{2})^2 + 4(-\frac{9\sqrt2}{2}) - 4(\frac{9\sqrt2}{2}) =30.09}[/tex]
Hence, the extreme values of the function are: -8 and 131.91
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Write the equation of a line, in general form , that has an slope of zero and a y-intercept of (0, -6).
Find the length and width of a rectangle that has the given perimeter and a maximum area. perimeter: 128 meters
Find the difference: 16.25 - 7.92
five friends went to a local restaurant and all had the buffet at $6.75 each. the following is the restaurant bill. does it seem reasonable 5($6.75) =&67.50
Find the rate of change of the area of a square with respect to the length z , the diagonal of the square. what is the rate when z=2?
The rate of change of Area, A with respect to the diagonal length, z and the rate of change when z = 2 is :
[tex]\frac{dA}{dz} = z [/tex][tex]\frac{dA}{dz} = 2 \: when \: z = 2 [/tex]The area of a square is rated to its diagonal thus :
Area of square = ALength of diagonal = zThe relationship between Area and diagonal of a square is : [tex]A \: = 0.5 {z}^{2} [/tex]
The rate of change of area with respect to the length, z of the square's diagonal ;
This the first differential of Area with respect to z
[tex] \frac{dA}{dz} = 2(0.5)z \: = z[/tex]
Therefore, the rate of change of area, A with respect to the length, z of the diagonal is [tex]\frac{dA}{dz} = z [/tex]
The rate of change [tex]\frac{dA}{dz} [/tex] when z = 2 can be calculated thus :
Substitute z = 2 in the relation [tex]\frac{dA}{dz} = z [/tex]
Therefore, [tex]\frac{dA}{dz} = 2 [/tex]
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Toby exercises 14 hours a week. John exercises 20% more than Toby and Jenny exercises two more hours than John. Which expression represents how much Jenny exercises? (w represents weeks)
the answer is 18.8w
Toby 14w
John 14(1.2)w
Jenny 14(1.2)w + 2w
thus, Jenny exercise
14(1.2)w + 2w
16.8w + 2w
18.8w
In Miss Marshalls classroom 6/7 of the students play sports of the students who play sports for fifth also play instruments if there are 35 students in her class how many play sports and instruments
what is the least common denominator for 5/6 and 3/8
For the x-values 1, 2, 3, and so on, the y-values of a function form a geometric sequence that increases in value. What type of function is it? A. Exponential growth B. Decreasing linear C. Increasing linear D. Exponential decay
Answer:
Option A: Exponential growth
Step-by-step explanation:
If for values of x values of y increases and forms a geometric sequence then it would be an exponential growth function because geometric sequence is an exponential function and since, it is increasing hence, an exponential growth.
Option B is incorrect because because y is not decreasing
Option C and D are incorrect because geometric sequence can never be linear since, it gives common ratio.
Final answer:
The function described, where y-values form an increasing geometric sequence as x-values increase, is characterized as Exponential growth. This is consistent with an exponential curve where the rate of growth is proportional to the current value, which aligns with the behavior of a geometric sequence.
Explanation:
For the given scenario where the y-values of a function form a geometric sequence that increases for the x-values 1, 2, 3, and so on, the type of function is described as Exponential growth. This is because in a geometric sequence each term after the first is found by multiplying the previous term by a constant called the common ratio, which is analogous to the exponential function where the growth rate of the value is proportional to its current value. The more the x-value increases, the higher and faster the y-value grows, following the pattern of an exponential growth curve.
Answer A. Exponential growth fits the description as it involves an increasing geometric sequence. Answer B. Decreasing linear refers to a linear function that decreases with each increment in x, which does not describe the function in question. Similarly, Answer C. Increasing linear describes a function that increases at a constant rate, not geometrically. Answer D. Exponential decay would imply the y-values decrease as x increases, which is also not the case described.
PLEASE HELP 15 POINTS WILL GIVE BRAINLIEST The following formula, F = ma, relates three quantities: Force (F), mass (m), and acceleration (a). A: Solve this equation, F = ma for a. B If F = -24 units and m = 10 units, what is the acceleration, a? Use the equation from Part (a) to answer the question. C: If F = 24 units and a = 12 units, what is the mass, m? Use the equation, F = ma, to plug in the known values and solve for m
If s = {r, u, d} is a set of linearly dependent vectors. if x = 5r + u + d, determine whether t = {r, u, x} is a linearly dependent set
A car manufacturer claims that, when driven at a speed of 50 miles per hour on a highway, the mileage of a certain model follows a normal distribution with mean μ = 30 miles per gallon and standard deviation σ= 4 miles per gallon.
Eric has 4 bags of 10 marbles and 6 single marbles . how many marbles does eric have
Determine all possible angles θ for equilibrium, 0∘<θ<90∘.