Consider the function f(x) = 3x + 8x ^ - 1 For this function there are four important intervals: (- infty,A],[A,B),(B,C] and [C, infty) where A, and Care the critical numbers and the function is not defined at
![Consider The Function F(x) = 3x + 8x ^ - 1 For This Function There Are Four Important Intervals: (- Infty,A],[A,B),(B,C]](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/8q1mVHHmZX39Oc82e9du0XoDz4ffZzWI.jpeg)
Answers
Given the following function:
\(f(x)=3x+8x^{-1}\)We will find the critical points A and C and the point B at which the function is not defined
First, we will find B:
f(x) is a rational function
The domain will be the real numbers except for the zeros of the denominator
The denominator has a zero at x = 0
So, when x = B = 0, the function will not be defined
B = 0
Second, we will find A and C
We need to find the first derivative df/dx and then solve the equation df/dx=0
\(\frac{df}{dx}=3-8x^{-2}\)Now, solve the equation df/dx = 0
\(\begin{gathered} 3-8x^{-2}=0 \\ 3-\frac{8}{x^2}=0 \\ \\ \frac{8}{x^2}=3 \\ \\ x^2=\frac{8}{3}\to x=\pm\sqrt{\frac{8}{3}}=\pm\frac{2\sqrt{6}}{3} \end{gathered}\)So, the values of A and C will be as follows:
\(\begin{gathered} A=-\frac{2\sqrt{6}}{3} \\ \\ C=\frac{2\sqrt{6}}{3} \end{gathered}\)So, the answer will be:
\(\begin{gathered} A=-\frac{2\sqrt{6}}{3} \\ \\ B=0 \\ \\ C=\frac{2\sqrt{6}}{3} \end{gathered}\)( -∞, A) : Increasing
(A, B) : decreasing
(A, C) : decreasing
(C, ∞) : Increasing
(-∞, B): Concave down
(B, ∞): concave up
Related Questions
Point M is the midpoint of FG. Can the value of variable a be determined? Explain F(b+1, a+2). M(3,5). G(2a, 3b+3)
Answers
Answer:
YES, the value of a is 2Step-by-step explanation:
Given the coordinates F(b+1, a+2, M(3,5) and G(2a, 3b+3), if M is the midpoint of F and G then, to get the value of a and b we will use the formula for calculating the midpoint a shown;
M(X, Y) = (x₁+x₂/2, y₁+y₂/2)
X = x₁+x₂/2 and Y = y₁+y₂/2
From the given coordinates x₁ = b+1, y₁ = a+2, x₂ = 2a, y₂ = 3b+3, X = 3 and Y = 5
Substituting the given parameters into the formula, we can get the value of a and b as shown;
3 = b+1+2a/2
6 = 2a+b+1
2a+b = 5 ............... 1
Similarly, for the Y coordinate;
5 = a+2+3b+3/2
10 = a+2+3b+3
a+3b = 10-2-3
a+3b = 5 ........ 2
Solving equation 1 and 2 simulatneously using elimination method
2a+b = 5 ............... 1 * 1
a+3b = 5 ........ 2 * 2
---------------------------------------------------------
2a+b = 5 ...........3
2a +6b = 10.......... 4
subtract 3 from 4
b - 6b = 5-10
-5b = -5
b = -5/-5
b = 1
substitute b =1 into equation 2 to get a;
a + 3b = 5
a + 3(1) = 5
a = 5-3
a = 2
From the calculation above it is seen that the value of variable a can be determined and it is equivalent to 2
Answer:
To determine the value of a, set up and solve a system of equations in two variables. The value of a is 2
Step-by-step explanation:
The relation of a line graph is visualized by drawing
lines between all of the points on a grid.
Answers
The relation of a line graph is visualized by drawing lines between all of the points on a grid is false.
What is a line graph?A line graph is a type of graph that uses lines to connect data points. The points are typically plotted on a grid, but the lines do not need to connect all of the points. In fact, it is often more helpful to only connect the points that are relevant to the data that you are trying to visualize.
For example, when graphing temperature over time, there is only need to connect the points that represent the temperature at each hour. There is no need to connect the points that represent the temperature at midnight, 2am, 4am, etc.
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Complete question:
The relation of a line graph is visualized by drawing lines between all of the points on a grid. Is this true or false? Explain your answer.
5 Signs for science project displays are cut of poster board that measure 1 yard on each side. Each sign is-yard long and-yard wide. How ma signs can be cut from 1 piece of poster board? Wh the area of each sign? Show your work.
Answers
Answer:
\(\text{27}\)
Step-by-step explanation:
Given that :
\(\text{Dimension of poster board} = 1 \ \text{yd} \ \text{by} \ 1 \ \text{yd}\)
\(\text{Dimension of each poster board} = \dfrac{1}{3} \ \text{yd} \ \text{by} \ \dfrac{1}{9} \ \text{yd}\)
Number of poster signs that can be cut :
\(\text{Area of poster sign} = \dfrac{1}{3} \times \dfrac{1}{9} = \dfrac{1}{27} \ \text{yard}^2\)
\(\text{Area of poster board} = 1 \ \text{yard}^2\)
Number of poster signs that can be cut :
\(\dfrac{\text{Area of poster board}}{\text{Area of poster sign}}\)
\(1 \ \text{yard}^2\div (\dfrac{1}{27} ) \ \text{yard}^2\)
\(1 \div \dfrac{1}{27}\)
\(1 \times \dfrac{27}{1}\)
\(\bold{= 27 \ poster \ signs}\)
Triangle FGH, with vertices F(-5,-7), G(-2,-5), and H(-6,-2), is drawn inside a rectangle, as shown below.
Answers
The area of the triangle FGH is equal to 13√10 unit²
Given that the vertices;
F(-5,-7), G(-2,-5), and H(-6,-2)
We have to find the area of the triangle as;
Area of triangle = 1/2bh²
Here,
Area of triangle FGH = 1/2 (GH) (FG)²
Now,
Length of FG = √26
Length of GH = √10
Then,
Area of triangle FGH = 1/2 (GH) (FG)²
Area of triangle = 1/2 × √10 × √26²
Area of triangle = 1/2 × √10 × 26
Area of triangle FGH = 13√10
Therefore,
The area of the triangle FGH = 13√10 unit²
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2 +2 pls give me the answer
Answers
Answer:
4
Step-by-step explanation:
2+2=4
I didn't know how to do this "Who, if anyone, traveled at a aster speed? Explain."
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Given:
The graph shows the relationship between the total distance Dominique traveled and the time.
The distance Ella traveled after x hours is represented by the equation,
\(y=550x..........\left(1\right)\)To find out: who traveled at a faster speed if anyone.
Explanation:
Let us find the equation of the line for Dominique.
Consider two points,
\(\lparen0,0),\left(2,1100\right)\)Using the two-point formula,
\(\begin{gathered} \frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1} \\ \frac{y-0}{1100-0}=\frac{x-0}{2-0} \\ \frac{y}{1100}=\frac{x}{2} \\ y=\frac{1100}{2}x \\ y=550x..........\left(2\right) \end{gathered}\)It is of the form,
\(y=mx+c,\text{ where m is the slope.}\)Since the slope (i.e. speed) of the equation is m = 550 which is equivalent to the slope (i.e speed) of the equation Ella's.
Therefore, both traveled at the same speed.
Final answer:
Both traveled at the same speed.
Each side length of a triangle is 4 cm. What type of triangle is it?
☐ right
□acute
Equilateral
Isosceles
Answers
Answer: Equilateral
Step-by-step explanation:
It's equilateral because it says that each side of the triangle is 4cm. In simple words, all the sides of triangle are equal. Such a triangle is called an equilateral triangle.
Hope you understood.
As part of a larger experiment, Dale (1992) looked at six samples of a wetland soil undergoing a simulated snowmelt. Three were randomly selected for treatment with a neutral pH snowmelt; the other three got a reduced pH snowmelt. The observed response was the number of Copepoda removed from each microcosm during the first 14 days of snowmelt Reduced pH Neutral pH 256 159 149 54 123 248 (a) [3 points Write two simple models: one model for the number of Copepoda in a neutral pH snowmelt and one model for the number of Copepoda in a reduced pH snowmel. Use mathematical notations. (b) 2 points We want to test whether the two treatments have equal average numbers of Copepoda. Write the n and the alternative hypotheses using the notations defined in the previous question (c) 5 points We want to perform a t-test. What assumptions are you making? Compute the corresponding test statistic. Give the critical value for a level of significance v = 0.05. What do you conclude? (d) 5 points) Using a randomization-based analysis, test the nul hypothesis that the two treatments have equal average numbers of Copepoda versus a two-sided alternative. Use a level of significance v = 0.05.
Answers
(a) The mathematical notations is (xₙ, yₙ)
(b) The alternative hypotheses using the notations defined in the previous question is t = (Xₙ - Yₙ) / SE
(c) The assumptions making is variances of the two populations are equal.
A hypothesis is a tentative explanation or prediction of a phenomenon, and it is essential in scientific research. In this study, we have two hypotheses:
Null Hypothesis: There is no significant difference in the average number of Copepod between the reduced pH and neutral pH snowmelt treatments.
Alternative Hypothesis: There is a significant difference in the average number of Copepod between the reduced pH and neutral pH snowmelt treatments.
Model:
A model is a mathematical representation of a phenomenon, and it is used to make predictions or explanations. In this experiment, we can create two simple models:
Model for Neutral pH Snowmelt:
Let Yₙ be the number of Copepod in a microcosm treated with neutral pH snowmelt. Then, Yₙ follows a normal distribution with mean µn and standard deviation σn.
Model for Reduced pH Snowmelt:
Let Yₙ be the number of Copepod in a microcosm treated with reduced pH snowmelt. Then, Yₙ follows a normal distribution with mean µr and standard deviation σr.
Assumptions:
Before conducting a t-test, we must ensure that the data satisfies the following assumptions:
The sample is a random and representative sample of the population.
The data is normally distributed.
The variances of the two populations are equal.
The test statistic for the t-test is:
t = (Xₙ - Yₙ) / SE
where Xₙ is the sample mean of neutral pH snowmelt, Yₙ is the sample mean of reduced pH snowmelt, and SE is the standard error.
Assuming a level of significance v = 0.05, we have a critical value of t = 2.306 for a two-tailed test with 4 degrees of freedom. If our calculated t-value is greater than 2.306 or less than -2.306, we reject the null hypothesis.
We can perform a randomization test as follows:
If the p-value is less than 0.05, we reject the null hypothesis and conclude that there is a significant difference in the average number of Cope pod between the reduced pH and neutral pH snowmelt treatments. If the p-value is greater than 0.05, we fail to reject the null hypothesis.
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2. What is the quotient of 6 and -1/2?
O -12
O -1/12
Answers
Answer:
- 12
hope this helps you!
Each individual letter in TENNESSEE is placed on
a piece of paper in a hat. If one letter is selected at
random from a hat, determine probability that:
15. Letter S is selected
16. A consonant is selected
17. Letter T or N is selected
18. The letter V is not selected
19. The letter W is selected
20. The letter S is not selected
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16. 5/9
17. 3/9
18. 9/9
19. 0/9
20. 7/9
tim has two spinners
Answers
PLEASE HELP ME ON QUESTION ONE ASAP!!
Answers
Answer:
The answer is D. 56
Step-by-step explanation:
Answer:
Oh yeah, super easy. 14
Step-by-step explanation: For every 6 mice there is 2 snakes, You can simplify to for every 3 mice there is 1 snake. All you need to do is divide 42 by 3 which would give you 14!
What is the solution to the compound inequality: 10<2x-4<20
Answers
Answer:
7 < x < 12
Step-by-step explanation:
To solve this inequality, solve it the same way you do a normal function. Perform the same actions on both sides of the inequality.
\(10 < 2x-4 < 20\\14 < 2x < 24\\7 < x < 12\)
A manufacturer knows that their items have a normally distributed lifespan, with a mean of 13.9 years, and standard deviation of 2.9 years. If you randomly purchase 9 items, what is the probability that their mean life will be longer than 14 years
Answers
Answer:
0.4588
Step-by-step explanation:
When we have random sample, the z score formula we use is:
z = (x-μ)/σ/√n
where x is the raw score = 14
μ is the population mean = 13.9
σ is the population standard deviation = 2.9
n is random number of samples = 9
z = 14 - 13.9/2.9/√9
z = 0.1/2.9/3
z = 0.10345
Probability value from Z-Table:
P(x<14) = 0.5412
P(x>14) = 1 - P(x<14) = 0.4588
The probability that their mean life will be longer than 14 years is 0.4588
(i) Write the zeroes of the polynomial by using above graph.
(ii)Form a quadratic polynomial for above graph.
(iii)If a,1/a are the zeroes of polynomial 2x² -x +8k, then find the value of k.
please answer
no spam only for darkparadox #darkparadox
Answers
There is no real Value of k that will satisfy the equation 2x² - x + 8k = 0 if a and 1/a are the roots of the polynomial.
(i) Zeroes of the polynomial:
In the graph, we have two points where the curve intersects the x-axis: one is at (-1,0), and the other is at (2,0).The corresponding values of x are -1 and 2, and they are the zeros of the polynomial. Therefore, the zeros of the polynomial are -1 and 2.(ii) Forming the quadratic polynomial:
From the graph, we can observe that the curve intersects the y-axis at the point (0,5), implying that the constant term of the polynomial is 5.
We can use the formula to find the quadratic polynomial if we have two zeros and one constant term. Thus, the quadratic polynomial is given by:(x + 1)(x - 2) = x² - x - 2x + 2 = x² - 3x + 2. Therefore, the quadratic polynomial is x² - 3x + 2.(iii) Value of k if a, 1/a are the zeroes of the polynomial 2x² - x + 8k:
We know that a and 1/a are the zeroes of the polynomial 2x² - x + 8k. Therefore, we can find the sum and product of the roots and use them to determine the value of k.
The sum of the roots is a + 1/a, and their product is a(1/a) = 1. Using the sum and product of the roots, we can write: a + 1/a = 1/2 (1/2 is the coefficient of x)Substituting a with 1/a in the above equation, we get: 1/a + a = 1/2Multiplying both sides of the equation by 2a, we get: 2 + 2a² = a
Simplifying the equation, we get: 2a² - a + 2 = 0Multiplying both sides by 2,
we get: 4a² - 2a + 4 = 0Dividing both sides by 2, we get: 2a² - a + 2 = 0
Using the quadratic formula, we get: a = [1 ± √(1 - 4(2)(2))]/(2(2))
Simplifying, we get: a = [1 ± √(-31)]/4Since the discriminant of the quadratic formula is negative, the roots are imaginary. Therefore, there is no real value of k that will satisfy the equation 2x² - x + 8k = 0 if a and 1/a are the roots of the polynomial.
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I need help with this question please using GeoGebra
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The 90% confidence interval for the mean is given as follows:
\(5.8 < \mu < 7.8\)
What is a t-distribution confidence interval?The t-distribution is used when the standard deviation for the population is not known, and the bounds of the confidence interval are given according to the following rule:
\(\overline{x} \pm t\frac{s}{\sqrt{n}}\)
In which the variables of the equation are presented as follows:
\(\overline{x}\) is the sample mean.t is the critical value.n is the sample size.s is the standard deviation for the sample.The critical value, using a t-distribution calculator, for a two-tailed 90% confidence interval, with 41 - 1 = 40 df, is t = 1.6839.
The parameters are given as follows:
\(\overline{x} = 6.8, s = 3.7, n = 41\)
The lower bound of the interval is given as follows:
6.8 - 1.6839 x 3.7/sqrt(41) = 5.8.
The upper bound of the interval is given as follows:
6.8 + 1.6839 x 3.7/sqrt(41) = 7.8.
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Please someone help me!
Answers
9514 1404 393
Answer:
P = (8, 0)
Step-by-step explanation:
The coordinates of point P are the weighted average of the end points. The weights are the reverse of the segment divisions.
P = (2/5)A +(3/5)B
P = (2(-4, -6) +3(16, 4))/5 = (-8+48, -12 +12)/5 = (40, 0)/5
P = (8, 0)
1) Convert 2-7i to trigonometric form
2) Use the n-th roots theorem to find the requested roots of the given complex number.
Find the cube roots of 125
Answers
Answer:
1) \(\sqrt{53}(\cos286^\circ+i\sin286^\circ)\)
2) \(\displaystyle 5,-\frac{5}{2}+\frac{5\sqrt{3}}{2}i,-\frac{5}{2}-\frac{5\sqrt{3}}{2}i\)
Step-by-step explanation:
Problem 1
\(z=2-7i\\\\r=\sqrt{a^2+b^2}=\sqrt{2^2+(-7)^2}=\sqrt{4+49}=\sqrt{53}\\\\\theta=\tan^{-1}(\frac{y}{x})=\tan^{-1}(\frac{-7}{2})\approx-74^\circ=360^\circ-74^\circ=286^\circ\\\\z=r\,(\cos\theta+i\sin\theta)=\sqrt{53}(\cos286^\circ+i\sin 286^\circ)\)
Problem 2
\(\displaystyle z^\frac{1}{n}=r^\frac{1}{n}\biggr[\text{cis}\biggr(\frac{\theta+2k\pi}{n}\biggr)\biggr]\,\,\,\,\,\,\,k=0,1,2,3,\,...\,,n-1\\\\z^\frac{1}{3}=125^\frac{1}{3}\biggr[\text{cis}\biggr(\frac{0+2(2)\pi}{3}\biggr)\biggr]=5\,\text{cis}\biggr(\frac{4\pi}{3}\biggr)=5\biggr(-\frac{1}{2}-\frac{\sqrt{3}}{2}i\biggr)=-\frac{5}{2}-\frac{5\sqrt{3}}{2}i\)
\(\displaystyle z^\frac{1}{3}=125^\frac{1}{3}\biggr[\text{cis}\biggr(\frac{0+2(1)\pi}{3}\biggr)\biggr]=5\,\text{cis}\biggr(\frac{2\pi}{3}\biggr)=5\biggr(-\frac{1}{2}+\frac{\sqrt{3}}{2}i\biggr)=-\frac{5}{2}+\frac{5\sqrt{3}}{2}i\)
\(\displaystyle z^\frac{1}{3}=125^\frac{1}{3}\biggr[\text{cis}\biggr(\frac{0+2(0)\pi}{3}\biggr)\biggr]=5\,\text{cis}(0)=5(1+0i)=5\)
Note that \(\text{cis}\,\theta=\cos\theta+i\sin\theta\) and \(125=125(\cos0^\circ+i\sin0^\circ)\)
Consider the phrase "the sum of a number and 8." What mathematical expression represents this phrase?
Answers
Answer:
x + 8
Step-by-step explanation:
Let the missing number be x
According to question,
x + 8
2(x + 4)Evaluate5x – 6 when x = 2.
Answers
Solve 12−5x−4kx=y for x.
Answers
Answer:
x = \(\frac{12-y}{4k+5}\)
Step-by-step explanation:
12-5x-4kx=y
1. Subtract 12 from both sides.
2.Distrubutive property (factor) a(b+c)=ab+ac
x(-5-4k)=y-12
Divide by -5-4kx = \(\frac{y-12}{-5-4k}\)
= \(\frac{12-y}{4k+5}\)
Find the value of x.
78°
OA7
OB. 8
O C. 5
O D. 1
(12x+18)
Answers
Step-by-step explanation:
you can find the answer by vertical opposite angle
if the value of x is computed and it's equal to 78 you will find the answer
78=(12x+18)^
78=(12×5+18)
78=(60+18)
78=78 the problem is solved
may u get branliest please please
Pleaseee help meee
The sum of three lengths of a fence ranges from 45 to 60 inches. Two side lengths are 9 and 18 inches If the wngth of the third siden xcess compound inequality to show the possible lengths of the third side
Answers
Answer:
45 < 9 + 18 + x < 60
45 < 27 + x < 60
18 < x < 33
PLEASE HELP I HAVE A TIME LIMIT ON THIS!!
Answers
If y is not on its own, then the coefficient (the number before the letter) of x should be 4 times bigger than the coefficient of y
A) Find an equation for the line perpendicular to the tangent line to the curve y=x^3-4x+6 at the point (2,6)
-The equation is y=
b) What is the smallest slope on the curve? At what point on the curve does the curve have this slope?
-The smallest slope on the curve is
-The curve has the smallest slope at the point
c) Find equations for the tangent lines to the curve at the points where the slope of the curve is 8.
Answers
Answer:
f(x) = x³ - 4x + 6
f'(x) = 3x² - 4
a) f'(2) = 3(2²) - 4 = 12 - 4 = 8
6 = 8(2) + b
6 = 16 + b
b = -10
y = 8x - 10
b) 3x² - 4 = 0
3x² = 4, so x = ±2/√3 = ±(2/3)√3
= ±1.1547
f(-(2/3)√3) = 9.0792
f((2/3)√3) = 2.9208
c) 3x² - 4 = 8
3x² = 12
x² = 4, so x = ±2
f(-2) = (-2)³ - 4(-2) + 6 = -8 + 8 + 6 = 6
6 = -2(8) + b
6 = -16 + b
b = 22
y = 8x + 22
f(2) = 6
y = 8x - 10
The equation perpendicular to the tangent is y = -1/8x + 25/4
-The smallest slope on the curve is 2.92
The curve has the smallest slope at the point (1.15, 2.92)
The equations at tangent points are y = 8x + 16 and y = 8x - 16
Finding the equation perpendicular to the tangentFrom the question, we have the following parameters that can be used in our computation:
y = x³ - 4x + 6
Differentiate
So, we have
f'(x) = 3x² - 4
The point is (2, 6)
So, we have
f'(2) = 3(2)² - 4
f'(2) = 8
The slope of the perpendicular line is
Slope = -1/8
So, we have
y = -1/8(x - 2) + 6
y = -1/8x + 25/4
The smallest slope on the curveWe have
f'(x) = 3x² - 4
Set to 0
3x² - 4 = 0
Solve for x
x = √[4/3]
x = 1.15
So, we have
Smallest slope = (√[4/3])³ - 4(√[4/3]) + 6
Smallest slope = 2.92
So, the smallest slope is 2.92 at (1.15, 2.92)
The equation of the tangent lineHere, we set f'(x) to 8
3x² - 4 = 8
Solve for x
x = ±2
Calculate y at x = ±2
y = (-2)³ - 4(-2) + 6 = 6: (-2, 0)
y = (2)³ - 4(2) + 6 = 6: (2, 0)
The equations at these points are
y = 8x + 16
y = 8x - 16
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When the sum of x and 4 is multiplied by 2,the product is 14,find x.
Answers
Answer:
x = 3Step-by-step explanation:
According to the question, the equation is:
(x + 4) × 2 = 14x + 4 = 14 / 2x + 4 = 7x = 7 - 4x = 3Let's see
2(x+4)=14x+4=14/2x+4=7x=7-4x=3the table shows the total cost of bowling any number games and renting bowling shoes. write a two step equation to represent the total cost of c for bowling g games.
Answers
Thus, an equation to show the overall price y of renting shoes and bowling x equipment is: y=4x+b.
Explain about the slope intercept form:The task is to create an equation that use the total expense (y) and the total number of bowled games (x). Hence, the way that we will begin is by entering the knowing into the formula of slope intercept form: y=mx+b.
The cost of a game is $4, while the cost of renting shoes is set and unknowable. Since each game costs $4, we already have. Hence, y=4x+b. Yet b is still unknown to us. The constant b in the equation y=mx+b usually refers to the fixed price in linear math equations.We are aware of the a y, overall game cost, and the x, overall game number. These figures are, respectively, $14 and 3.Thus, we include them in the formula: y=4x+b.
14 = 4(3)+b
Find b
14 = 12+b
14 - 12=b
b = 2
Thus, an equation to show the overall price y of renting shoes and bowling x equipment is: y=4x+b.
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Complete question:
The total cost for bowling includes the fee for shoe rental plus a fee per game. The cost of the game increases the price by $4. After 3 games, the total cost with shoe rental is $14.
a.write an equation to represent the total cost y to rent shoes and bowl x games.
The sum of two numbers and b is equal to c The number line below shows the values of a and c. What is the value of b?
Answers
Answer:
1
Look how a and c is spaced out. The, find the middle. Then, just like that, you see the middle. It's 1!
bella answered 33 questions on her math test. she earned a 92% on the test. how many questions did she answer correct
Answers
Answer: 30.36
Step-by-step explanation:
92% of 33 is 30.36. I don't know how Bella got 30.36 questions correct, but that is what I got.
What is the area of the base of the cone below? 12ft v=90ft^3
Answers
Answer: C 22.5 ft2!
Step-by-step explanation:
i took the test, have a great day:) <3
Answer:
right
Step-by-step explanation: