Suppose that y varies directly with x and y = 2 when x =16 write a direct variation equation that relates x and y
The equation of the direct variation is expressed as: y = 1/8x.
How to Write a Direct Variation Equation?The equation of a direct variation between two variables, say x and y, is expressed as y = kx, where k is the constant of proportionality, if y varies directly as x.
Therefore, substitute y = 2 and x = 16 into y = kx to find the value of k:
2 = k(16)
Divide both sides by 16
2/16 = k
1/8 = k
k = 1/8.
To write the equation of the direct variation, substitute k = 1/8 into y = kx:
y = 1/8x.
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Helpppppp plas I don’t know the answer and I’m crying
The given expressions are:
A: 3(x+2)
B: 3x+6
When two expressions are equivalent, it doesn't matter which x-value you use, the result will always be the same, then option C and option D show that these expressions are equivalent, because we will obtain the same result, regardless of the value of x.
Now, the distributive property states:
a(x+y)=a*x+a*y
If we apply this property to expression A, we have:
3*x+3*2=3x+6
Thus, by applying the distributive property we can see that those expressions are equivalent.
The option B which says "Both expressions involve addition" does not show that these expressions are equivalent since we can have different expressions as: 4x+8, x+1, etc... and these are not equivalent.
Thus, the statement which doesn't show that these expressions are equivalent is B. Both expression involve addition..
Now, the
translate the following into an equation:five increased by triple a number results in 52
Answer:
3n+5=52
Explanation:
• Let the number = n
,• Triple the number = 3n
,• Five increased by triple a number = 3n+5
,• The result is 52.
The equation is:
[tex]3n+5=52[/tex]Complete the square to rewrite the equation into standard form, show all work please x^2+6x+y^2-4y=-9
1) The first step is to divide the coefficient b by 2
[tex]\begin{gathered} b=6\therefore\frac{6}{2}=3 \\ b=-\frac{4}{2}\therefore=-\frac{4}{2}=-2 \end{gathered}[/tex]2) So now, let's add to both sides those squared numbers to both sides of the equation and then write them as binomials:
[tex]\begin{gathered} x^2+6x+3^2+y^2-4x+(-2)^2=-9+9+4 \\ (x+3)^2+(y-2)^2=4 \end{gathered}[/tex]Note that this is the Equation of the Circle in its standard form, whose radius is 4
Y is inversely proportional to the cube of x. If Y = 5 when x = 2, then k = 20.
Answer:
False
Explanation:
Y is inversely proportional to the cube of x. Mathematically, this means
[tex]y\propto\frac{1}{x^3}[/tex]If we now introduce a proportionality constant k, then we get
[tex]y=\frac{k}{x^3}[/tex]Now if y = 5 when x = 2, then
[tex]5=\frac{k}{2^3}[/tex][tex]5=\frac{k}{8}[/tex]Multiplying both sides by 8 gives
[tex]5\times8=k[/tex][tex]\boxed{k=40.}[/tex]Hence, the value of k is NOT 20.
Therefore, the statement that "If Y = 5 when x = 2, then k = 20." is false.
f(x) = 6x^4 + 6Use the limit process to find the slope of the line tangent to the graph of f at x = 2. Slope at x= 2:__Find an equation of the line tangent to the graph of f at x = 2:__
The given function is
f(x) = 6x^4 + 6
The formula for the limit is shown below
[tex]\begin{gathered} f^{\prime}(x)\text{ = }\lim _{h\to0}\text{ }\frac{f(x\text{ + h) - f(x)}}{h} \\ \text{Substituting x = x + h into the function, we have} \\ f^{\prime}(x)\text{ = }\lim _{h\to0}\text{ }\frac{6(x+h)^4+6-(6x^4+6)}{h} \\ f^{\prime}(x)\text{ = }\lim _{h\to0}\text{ }\frac{6(h^4+4h^3x+6h^2x^2+4hx^3+x^4)+6-6x^4-6}{h} \\ f^{\prime}(x)\text{ = }\lim _{h\to0}\text{ }\frac{6h^4+24h^3x+36h^2x^2+24hx^3+6x^{4\text{ }}-6x^4\text{ + 6 - 6}}{h} \\ f^{\prime}(x)\text{ = }\lim _{h\to0}\text{ }\frac{h(6h^3+24h^2x+36hx^2+24x^3)}{h} \\ h\text{ cancels out} \\ \end{gathered}[/tex]Evaluating the limit at h = 0, we would substitute h = 0 into 6h^3 + 24h^2x + 36hx^2 + 24x^3
It becomes
6(0)^3 + 24(0)^2x + 36(0)x^2 + 24x^3
The derivative is 24x^3
f'(x) = 24x^3
This is the slope of the tangent line is at x = 2
By substituting x = 2 into f'(x) = 24x^3, it becomes
f'(2) = 24(2)^3 = 192
To find the y coordinate of the point, we would substitute x = 2 into
f(x) = 6x^4 + 6
y = 6(2)^4 + 6 = 102
Thus, the x and y coordinates are (2, 102) and the slope is 192
The equation of the line in the point slope form is
y - y1 = m(x - x1)
Thus, the equation of the tangent is
y - 102 = 192(x - 2)
what is the x intercept?
Answer:
the x intercept is -2
Step-by-step explanation:
Banana Cupcakes
makes 10 cupcakes
1 cup granulated sugar
cup vegetable oil
1 large egg
4 tablespoons sour cream
2 medium-sized ripe bananas, mashed
1 cups all-purpose flour
1 teaspoon baking soda
teaspoon salt
1 teaspoon vanilla extract
pinch of nutmeg
You will use the recipe above to answer the following questions:
This recipe serves 10, but you need to serve 30. What number will you need to multiply the amount of each ingredient by to adjust the recipe? How did you determine this number?
How much vegetable oil do you need for 30 cupcakes?
How much flour do you need for 30 cupcakes?
What is the difference in the amount of vanilla extract you would need for 30 cupcakes?
What is the difference in the amount of salt you would need for 30 cupcakes?
In the real world, even though you make adjustments to a recipe to accommodate the number of people you need to serve, you sometimes round the amount of an ingredient instead of using an exact amount. Which ingredient would it make more sense to round rather than coming up with the exact amount? Why?
For 30 cupcakes we need 3 cups of vegetable oil, and 3 cups of flour, for 30 cupcakes we need 3 teaspoons of vanilla, the difference is 3 - 1 = 2.
Here we have a recipe that works for 10 serves.
Then, if we want to make 30 servings, we need to use that recipe 3 times. (because x*10 = 30, then x = 30/10 = 3)
This would be equivalent to multiplying each ingredient by 3.
for 30 cupcakes we need:
3 cups of vegetable oil.
3 cups of flour.
(because for each one of these ingredients we needed only 1 cup to make 10 cupcakes).
for 10 cupcakes we need 1 teaspoon of vanilla.
for 30 cupcakes we need 3 teaspoons of vanilla, the difference is 3 - 1 = 2.
So the difference is 2 teaspoons.
Hence, for 30 cupcakes we need 3 cups of vegetable oil, and 3 cups of flour, for 30 cupcakes we need 3 teaspoons of vanilla, the difference is 3 - 1 = 2.
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Fun zone offers four party packages. The prices for each package are $50 more on weekends than they are on weekdays, as shown in the table below
Okay, here we have this:
Considering the provided information, if the relationship between the number of packages and price is a function, so we obtain the following:
Since a function is a relation where each element of the first set is assigned a single element of the second set or none at all, in this case the first set is the number of the package, and the second the price. However, it does not comply with the requirement that each element of the first set is assigned a single element of the second.
Then, finally we obtain that since the condition is not satisfied, it means that it is not a set. And the correct answer will be the third option.
Help on any of these problems would be appreciated. Thanks! Question 1
Theorem: The measure of the angle at the center is equal to the measure of the angle at the circumference.
Hence, the answer is
[tex]x=70^0[/tex]The answer wasn't clear for me
Find the number of 5-digit numbers that contain only digits 1, 2, 3, 4, 5, and use each digit exactly once. the digits 1 and 2 cannot be neighbors. the digits 3 and 4 cannot be neighbors.
The number of 5 - digit number that contain only digits 1, 2, 3, 4, 5, and use each digit exactly once will be;
⇒ 35241
What is an expression?
Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The number of 5-digit numbers that contain only digits 1, 2, 3, 4, 5.
And, Use each digit exactly once. the digits 1 and 2 cannot be neighbors. the digits 3 and 4 cannot be neighbors.
Now,
The number of 5-digit numbers that contain only digits 1, 2, 3, 4, 5 and the digits 1 and 2 cannot be neighbors. the digits 3 and 4 cannot be neighbors will be;
⇒ 35241
Thus, The number of 5 - digit number that contain only digits 1, 2, 3, 4, 5, and use each digit exactly once will be;
⇒ 35241
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Find the area between the graph of y= -12x^3 and the x-axis on the interval [-1, 1]. Write the exact answer. Do not round.
Recall that the integral of the area between the graph of two functions, in an interval [a,b] is:
[tex]\int ^b_a|f(x)-h(x)|dx\text{.}[/tex]Now, if f(x) is an odd function, we can use the following property:
[tex]\int ^a_{-a}|f(x)|dx=2\int ^a_0|f(x)|dx\text{.}[/tex]Now, notice that the function y=-12x³ is an odd function, therefore:
[tex]\int ^1_{-1}|y-0|dx=2\int ^1_0|-12x^3|dx=2\int ^1_012x^3dx\text{.}[/tex]Applying the linearity of the integral we get:
[tex]24\int ^1_0x^3dx=24\frac{x^4}{4}|^1_0=24(\frac{1}{4}-0)=\frac{24}{4}=6.[/tex]Answer: 6.
the graph shows function m, a transformation of f(x) = x^1/2replace a and h to create the equation for function m. (picture of graph listed below)
Answer:
[tex]\begin{gathered} a\text{ = 2, h = 3} \\ m(x)\text{ =2\lparen x + 3\rparen}^{\frac{1}{2}} \end{gathered}[/tex]Explanation:[tex]\begin{gathered} The\text{ parent function:} \\ f(x)\text{ = x}^{\frac{1}{2}} \\ \\ new\text{ function is given as:} \\ m(x)\text{ = a\lparen x + h\rparen}^{\frac{1}{2}} \\ We\text{ need to find a and h} \end{gathered}[/tex]We need to translate the parent function in order to get the function for m(x).
The first thing we will do is to get the graph of the parent function. Then we will compare with the graph of m(x) to determine the transformation applied to it.
The parent graph starts at the origin (0, 0). But the graph of m(x) starts at -3. This means the graph was translated (moved) 3 units to the left.
When we move to the left, the value will be a positive number when writing the function
So our h = 3
[tex]\text{m\lparen x\rparen= a\lparen x + 3\rparen}^{\frac{1}{2}}[/tex]Next, is to find a:
The value of a could be less than 1, 1 or greater than 1. To determine the value, pick a point from the graph of m(x) given.
Using point (6, 6), we will substitute this value into the translated function to get a
[tex]\begin{gathered} \text{x = 6, y = 6} \\ y\text{ in this case = m\lparen x\rparen} \\ m(x)\text{ = a\lparen x + 3\rparen}^{\frac{1}{2}} \\ 6\text{ = a\lparen6 + 3\rparen}^{\frac{1}{2}} \\ 6\text{ = a\lparen9\rparen}^{\frac{1}{2}} \\ 6\text{ = a\lparen}\sqrt{9}) \\ 6\text{ = a\lparen3\rparen} \\ a\text{ = 6/3 } \\ \text{a = 2} \end{gathered}[/tex]To ascertain we got the right function, I'll plot with the values of h and a we got and compare:
[tex]\begin{gathered} a\text{ = 2, h = 3} \\ m(x)\text{ =2\lparen x + 3\rparen}^{\frac{1}{2}} \end{gathered}[/tex]The graph above is the same as the given one in the question
Write an equation in standard form of the line passing through the points (12, 6) and (-4, 10).The equation is . (Type your answer in standard form.)
Answer:
The equation, in the standard form is: x + 4y = 36
Step-by-step explanation:
The standard form of the equation of a line has the following format:
Ax + By = C.
First, I will place the equation of the line in slope-intercept formula, which is:
y = ax + b. Then, I pass to the standard.
Passes through the point (12,6):
This means that when x = 12, y = 6.
So
y = ax + b
6 = 12a + b
b = 6 - 12a
Passes through the point (-4,10):
This means that when x = -4, y = 10. SO
10 = -4a + b
Since b = 6 - 12a
10 = -4a + (6 - 12a)
10 = -4a + 6 - 12a
10 - 6 = -4a - 12a
-16a = 4
16a = -4
a = -4/16
Simplifying by 4
a = -1/4
b = 6 - 12a = 6 - 12*(-1/4) = 6 + 3 = 9
So
y = ax + b
y = -(x/4) + 9
(x/4) + y = 9
Multiplying everything by 4
x + 4y = 36
The equation, in the standard form is: x + 4y = 36
A vacant lot in the shape of a rectangle measures 110 feet by 30 feet A)what is the perimeter of the lot?B)what is the area of the lot?
SOLUTION
Since the vacant lot has the shape of a rectangle;
A. To measure the perimeter, we will use the formula for perimeter of a rectangle given as
[tex]P=2(l+b)[/tex]Where L is the length and b is the breadth or width
So, the perimeter becomes
[tex]\begin{gathered} P=2(l+b) \\ P=2(110+30) \\ P=2(140) \\ P=2\times140 \\ P=180ft \end{gathered}[/tex]B. The area of a rectangle is given as
[tex]A=l\times b[/tex]So, the area becomes
[tex]\begin{gathered} A=110\times30 \\ A=3300ft^2 \end{gathered}[/tex]Therefore, the Perimeter is 180 feet and the Area is 3300 square-feet.
Consider the linear equation 2y - 3x = 5.Are (-1, 1) and (4, 1) solutions to the inequality 2y - 3x < 5? Explain how you know.
Solution
For this case we have the following inequality:
2y-3x< 5
And we can solve for y like this:
2y < 3x+5
y < 1/2 (3x+5)
We can replace the points and we can verify:
x=-1 y=1/2*(3*-1 +5) = 1/2(-3+5)= 1 then y is not <1
x=4 y=1/2*(3*4 +5) = 1/2(12+5)= 17/2 then y is not <1
Nina has prepared the following two-column proof below. She is given that ZOLN = ZLNO andShe is trying to prove that ol=on
First error:
Draw OE as a perpendicular bisector of LN
First correction:
Draw OE as a bisector of By construction
Second error:
By transitive property of equality
x and y are equal because of the property of the sum of the internal angles of a triangle
a + b + x = 180°
a + b + y = 180°
Second correction:
Sum of the internal angles of a triangle
Noah has a coupon for 30% off at his favorite clothing store can you use it to buy a hoodie and a pair of jeans I paid $28 for the jeans after using the coupon what is the regular price
Given:
Coupon = 30%
Amount paid after using the coupon = $28
Let's find the regular price.
The coupon is a form of voucher that enables someone to get a discount off a product.
This means after a discount of 30%, the new price of the jeans is $28
Thus, to find the regular price, we have:
[tex]28=P(1-\frac{30}{100})[/tex]Where P represents the regular price.
From the equation above, let's solve for P.
[tex]28=P(1-0.3)[/tex][tex]\begin{gathered} 28=P(0.7) \\ \\ 28=0.7P \\ \\ \text{Divide both sides by 0.7:} \\ \frac{28}{0.7}=\frac{0.7P}{0.7} \\ \\ 40=P \\ \\ P=40 \end{gathered}[/tex]Therefore, the regular price for the Jeans is $40
ANSWER:
$40
On October 23, 2011, one U.S. dollar was worth 13.65 Mexican pesos.(a) On that date, how many pesos was 81.35 dollars worth?Round your answer to the nearest hundredth of a peso.pesos(b) On that date, how many dollars was 91.03 pesos worth?Round your answer to the nearest hundredth of a dollar.
Solution
We are given that
[tex]\text{ \$}1=13.65\text{ }pesos[/tex]a.
[tex]\begin{gathered} \text{ \$}81.35=81.35\times13.65\text{ }pesos \\ \\ \text{ \$}81.35=1110.43\text{ }pesos\text{ \lparen to the nearest hundredth\rparen} \end{gathered}[/tex]b.
[tex]\begin{gathered} 1\text{ }pesos=\text{ \$}\frac{1}{13.65} \\ \\ 91.03pesos=\text{ \$}\frac{1}{13.65}\times91.03 \\ \\ 91.03pesos=\text{ \$}6.67\text{ \lparen to the nearest hundredth\rparen} \end{gathered}[/tex]Question 9 Which equation would generate the arithmetic sequence: -5,- 14,- 23,-32,-41, A an = -5+9(n-1) B an = -5-9(n-1) C None of the other answers are correct D an = 9+5(n-1) E a = 9-5(n-1)
B
1) Examining the Arithmetic Sequence:
(-5,-14, -23,-32,-41,..)
We have the following information:
a_1 = -5
Common ratio:-9
2) From these data, we have the Explicit formula:
[tex]a_n=-5_{}-9(n-1)[/tex]3) Looking at the options we have some formulas, so we can state that the equation that would generate the Arithmetic Sequence is described in option B
Reasoning Krishan wants his quiz average to be at least90 so that he can get an A in the class. His current quiz scoresare: 80, 100, 85. What does he have to get on his 2052next quiz to have an average of 90?A 85B 90C 92D 95
Average = total of all test scores/number of tests
90 = ( 80 + 100 + 85 + x ) / 4
Solve for x
90 (4) = 80 + 100 +85 + x
360 = 265 + x
360-265 = x
95 = x
how long must $217 be invested at a rate of 6% to earn 78.12 dollars and interest
In ths case the answer is very simple.
Step 01:
Data:
p = investment = $217
r = interest rate = 6% = 0.06
I = interest = $78.12
t ( years) = Time = ?
Step 02:
Simple Interest
I = p * r * t
78.12 = 217 * 0.06 * t
[tex]\begin{gathered} \frac{78.12}{217\cdot0.06}=t \\ 6\text{ = t} \end{gathered}[/tex]The answer is:
t = 6 years
SHOW THE EQUATION YOU SET UP.91 is what percent of 14?
1) Given that we want to know how much in percentage is 91 to 14 we can write out:
[tex]\begin{gathered} 14-------100\% \\ 91--------x \\ 14x=9100 \\ \frac{14x}{14}=\frac{9100}{14} \\ x=650\% \end{gathered}[/tex]Note that we crossed multiply that. And since 91 is way over 14, then it is clearly greater than 100%
Thus, 91 corresponds to 650% of 14
A landscaper is designing a flower garden in the shape of a right triangle. She wants 10ft of a perennial border to form the hypotenuse of the triangle, and one leg is to be 2ft longer than the other. Find the lengths of the legs?
Let the length of one leg be x, then other leg length is (x+2).
Determine the length of leg of triangle by using the pythagoras theorem.
[tex]\begin{gathered} (10)^2=(x)^2+(x+2)^2 \\ 100=x^2+x^2+4x+4 \\ x^2+2x-48=0 \\ x^2+8x-6x-48=0 \\ x(x+8)-6(x+8)=0 \\ (x-6)(x+8)=0 \end{gathered}[/tex]So values of x is 6 and -8. Negative length is not possible. The value of x is 6.
Determine the length of other leg.
[tex]\begin{gathered} x+2=6+2 \\ =8 \end{gathered}[/tex]Thus length of legs of triangle are 6 ft and 8 ft.
For an equation with fractions,why is it helpful to multiply both sides of theequation by the LCD?
When you multiply both sides of an equation with fractions with the lowest common denominator, it simplifies the equation, so that adding/subtracting/multiplying and or dividing makes it easier.
Suppose that the speed at which cars go on the freeway is normally distributed with mean 73 mph and standard deviation 9 miles per hour. Let X be the speed for a randomly selected car. Round all answers to 4 decimal places where possible.a. What is the distribution of X? X ~ N(73Correct,9Correct) b. If one car is randomly chosen, find the probability that it is traveling more than 74 mph. 0.4558Correct c. If one of the cars is randomly chosen, find the probability that it is traveling between 72 and 76 mph. 0.17478Correct d. 69% of all cars travel at least how fast on the freeway? 68.5Incorrect mph.
Answer:
68.536 mph
Explanation:
Part D
• Mean Speed = 73 mph
,• Standard deviation = 9 mph.
Since we are supposed to find at least how fast on the freeway, then:
[tex]\begin{gathered} P(X\geq x)=0.69 \\ 1-P(XFrom the z-table, the z-value at 31% is -0.496.[tex]\begin{gathered} z=\frac{X-\mu}{\sigma} \\ -0.496=\frac{X-73}{9} \\ \text{ Cross multiply} \\ X-73=9\times-0.496 \\ X=73+(9\times-0.496) \\ X=68.536 \end{gathered}[/tex]69% of all cars travel at least 68.536 mph on the freeway.
By the congruent supplements theorem, what can youconclude?CBG = _DBGO_FBC = _DBG_CBG is supplementary to _DBF._FBC is supplementary to DBG.
The congruent supplements theorem basically states that if we have two pairs of supplementary angles, say A and B are supplementary and C and D are supplementary and one of angle of each pair are congruent, say A congruent to C, then the other two are also congruent (say B congruent to D)
In our case, the angle A is the angle FBC, B is the angle CBG. C is the angle DBG and D is the angle DBF. Since B is congruent to D then A is congruent to C. So angle FBC is congruent to angle DBG, which is option 2
what is the y- intercept when x =-1,0,1,2,3,4, and y= -4,-3.5,-3,-2.5,-2,-1.5
The y - intercept will be - 7/2.
What is Equation of line?
The equation of line passing through the points (x₁ , y₁) and (x₂, y₂) with slope m is defined as;
y - y₁ = m (x - x₁)
Where, m = (y₂ - y₁) / (x₂ - x₁)
Given that;
The values are,
x = -1, 0, 1, 2, 3, 4,
y = -4, -3.5, -3, -2.5, -2, -1.5
Now,
Take two points on the line as;
⇒ (x, y) = (- 1, - 4) , (1, - 3)
Hence, The slope of the line passing through the points (- 1, - 4) , (1, - 3) is;
⇒ m = (y₂ - y₁) / (x₂ - x₁)
⇒ m = (- 3 - (-4)) / (1 - (-1))
⇒ m = 1 / 2
So, The equation of line with slope 1/2 will be;
⇒ y - (-4) = 1/2 (x - (-1)
⇒ y + 4 = 1/2 (x + 1)
⇒ y + 4 = 1/2x + 1/2
⇒ y = 1/2x + 1/2 - 4
⇒ y = 1/2x - 7/2
Thus, The y - intercept will be - 7/2.
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Circle describe and correct each error Graph y=x-4 using slop-intercept form.M= -4y-int=1
Given:
Given that a graph of the function
[tex]\begin{gathered} y=x-4 \\ m=-4 \\ y-int=1 \end{gathered}[/tex]Required:
To find error in the given question.
Explanation:
The standard equation of the line is
[tex]y=mx+c[/tex]Where m slope and c is y-intercept.
Consider the given equation
[tex]y=x-4[/tex]Here the slope is 1 and y-intercept is at -4.
And the graph of the equation is,
Final Answer:
The error is :
[tex]\begin{gathered} m=-4 \\ y-int=1 \end{gathered}[/tex]Find the area of the figure. Use 3.14 for .18 in9 inO A. 97.2 in2O B. 122.24 in2O C. 61.12 in2D. 86.24 in2
SOLUTION
We want to solve the question below
The figure consists of a semi-circle and a triangle. So the area of the figure becomes
Area of semi-circle + area of triangle
The semi-circle has a diameter of 8 in. So the radius becomes
[tex]r=\frac{diameter}{2}=\frac{8}{2}=4in[/tex]Area of the semi-circle is given as
[tex]\begin{gathered} \frac{1}{2}\times\pi r^2 \\ \frac{1}{2}\times3.14\times4^2 \\ \frac{1}{2}\times3.14\times16 \\ =25.12\text{ in}^2 \end{gathered}[/tex]Area of the triangle is
[tex]\begin{gathered} \frac{1}{2}\times base\times height \\ \frac{1}{2}\times9\times8 \\ 9\times4 \\ =36\text{ in}^2 \end{gathered}[/tex]So Area of the figure becomes
[tex]\begin{gathered} 25.12+36 \\ =61.12\text{ in}^2 \end{gathered}[/tex]Hence the answer is option C