We are given slope and the y-intercept. The formula y = mx + c can be used to determine the equation of the line.
m = slope
c= y-intercept.
For this question,
m = 6
c = -2
The equation of the line is
y = 6x + (-2)
y = 6x -2
The answer is y = 6x -2
Two boats leave the same marina. One heads north, and the other heads
east. After some time, the northbound boat has traveled 39 kilometers, and
the eastbound boat has traveled 52 kilometers. How far apart are the two
boats
The distance travelled by the two boats forms a right triangle. Thus, applying the Pythagorean Theorem we find out that the two boats are 65 kilometers apart from each other after travelling 39 kilometers north and 52 kilometers east. Thus, 1st option is correct.
It is given to us that -
There are two boats
One boat heads north while the other heads east
The boat travelling north has traveled 39 kilometers
The boat travelling south has traveled 52 kilometers
We have to find out the distance between the two boats after they have travelled the respective distances.
It is known to us that one boat heads north while the other heads east. We can see that the trajectory formed between the two boats resembles a right triangle as they start from the same point.
One leg of the right triangle formed equals to the distance travelled by the boat travelling north.
Let us say the distance travelled by the boat travelling north be "a".
=> a = 39 kilometers ----- (1)
Other leg of the right triangle formed equals to the distance travelled by the boat travelling east.
Let us say the distance travelled by the boat travelling east be "b".
=> b = 52 kilometers ------ (2)
Now, the distance between the two boats after they have travelled the respective distances is equal to the value of the hypotenuse of the right triangle formed.
Let us say the hypotenuse of the right triangle formed be "h".
According to the Pythagorean Theorem for a right triangle,
[tex]a^{2} +b^{2} =h^{2}[/tex] ---- (3)
where, a, b = legs of the right triangle
and, h = hypotenuse of the right triangle
Substituting the values of a and b from equations (1) and (2) respectively in equation (3), we have
[tex]a^{2} +b^{2} =h^{2}\\= > 39^{2} +52^{2} =h^{2}\\= > h^{2}=1521+2704\\= > h^{2}=4225\\= > h=65[/tex]
So, the value of the hypotenuse of the right triangle formed is 65 kilometers.
Thus, applying the Pythagorean Theorem we find out that the two boats are 65 kilometers apart from each other after travelling 39 kilometers north and 52 kilometers east. Thus, 1st option is correct.
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−5x+8?I need the Answer please!!
We have the next expression
[tex]-5x+8[/tex]And we must solve it for x
[tex]undefined[/tex]Given the figure below, find the values of x and z. (9x + 70). (6x + 80).
( 9x+70)+(9x+70) + (6x + 80) + (6x+ 80 ) = 360
If you solve the equation you get that
x = --44/5
Now, since
z = ( 6x +80 ) = ( 6*(-44/5) + 80 ) = 136 / 5
So, there you have, x,z
finding the vertex, intercepts, and axis of symmetry from the graph of a parabola
Solution
Explanation:
Given:
(b) Equation of the axis of symmetry
[tex]\begin{gathered} x=-8 \\ x=4 \end{gathered}[/tex][tex]\begin{gathered} x=-8,x=4 \\ (x+8)(x-4)=0 \\ x^2-4x+8x-32=0 \\ x^2+4x-32=0 \\ y=x^2+4x-32 \end{gathered}[/tex]where
[tex]\begin{gathered} y=ax^2+bx+c \\ a=1,b=4,c=-32 \end{gathered}[/tex]The formula for the axis of symmetry and the x value of the vertex
[tex]x=-\frac{b^2}{2a}[/tex]Plug in the value
[tex]x=\frac{-(4)^}{2}=-2[/tex](d) To find the y value of the vertex, substitute 1 for x in the equation.
[tex]\begin{gathered} y=x^2+4x-32 \\ y=(-2)+4(-2)-32 \\ y=-2-8-32 \\ y=-42 \end{gathered}[/tex]The vertex is (-2 , -42) Since a > 0 the vertex is the minimum point and the parabola opens upward.
Hence the vertex = (-2 , -42)
The weight (W kg) of a decaying radio active substance after n years is given by W= Wo(1/2)^n/100, where Wo kg is the initial weight of the substance. 1. Atleast how many years will it take for the radioactive substance to lose to 10% of its initial weight?
Answer:
332.19 years
Explanation:
The weight, W of the substance after n years is given by:
[tex]W=W_o\mleft(\frac{1}{2}\mright)^{\frac{n}{100}}[/tex]Let the initial weight = 100%
If the substance loses to 10% of its initial weight, then:
• Wo = 100%
,• W= 10%
Substitute these into the formula:
[tex]\begin{gathered} \frac{10}{100}=\frac{100}{100}\mleft(\frac{1}{2}\mright)^{\frac{n}{100}} \\ \implies0.1=\mleft(\frac{1}{2}\mright)^{\frac{n}{100}} \end{gathered}[/tex]We then solve the equation for the value of n.
Take the logarithm of both sides.
[tex]\begin{gathered} \log (0.1)=\log \mleft(\frac{1}{2}\mright)^{\frac{n}{100}} \\ \implies\log (0.1)=\frac{n}{100}\log (\frac{1}{2})^{} \end{gathered}[/tex]Then divide both sides by log(1/2):
[tex]\begin{gathered} \frac{\log (0.1)}{\log (\frac{1}{2})}=\frac{\frac{n}{100}\log(\frac{1}{2})^{}}{\log(\frac{1}{2})} \\ \implies\frac{n}{100}=\frac{\log (0.1)}{\log (\frac{1}{2})} \end{gathered}[/tex]Finally, multiply both sides by 100:
[tex]\begin{gathered} 100\times\frac{n}{100}=100\times\frac{\log (0.1)}{\log (\frac{1}{2})} \\ n=332.19\text{ years} \end{gathered}[/tex]It will take at least 332.19 years for the radioactive substance to lose to 10% of its initial weight.
you are in a hot air balloon that is 600 feet above the ground. if the angle from your line of sight to your friend is 20°, how far is he from the point on the ground.
Answer
x = 164.9 ft
Explanation:
Given the following figures
To find the distance from the point on the ground, we need to apply the SOH CAH TOA
[tex]\begin{gathered} \text{Height = 600 ft} \\ \text{Horizontal distance x} \\ \theta\text{ = 20} \\ \text{ }\tan \theta\text{ =}\frac{opposite}{\text{adjacent}} \\ \text{opposite = 600 ft} \\ \text{adjacent = x ft} \\ \tan \text{ 20 = }\frac{600}{x} \\ \text{Cross multiply} \\ x\cdot\text{ tan 20 = 600} \\ \text{x = }\frac{600}{\tan \text{ 20}} \\ \tan \text{ 20 = }0.3639 \\ \text{x = }\frac{60}{0.3639} \\ \text{x = }164\text{ .9 ft} \end{gathered}[/tex]Therefore, the distance is 164.9 ft
Frank has a circular Garden the area of the garden is 100 ft Square what is the approximate distance from the edge of Frank's garden to the center of the garden (A= 3.14r ² )
ANSWER
[tex]5.64ft[/tex]EXPLANATION
The approximate distance from the edge of the garden to the center is the radius of the garden.
The area of a circle is given as:
[tex]A=\pi\cdot r^2[/tex]We can find the radius by making r the subject of the formula:
[tex]\begin{gathered} \frac{A}{\pi}=\frac{\pi\cdot r^2}{\pi} \\ r^2=\frac{A}{\pi} \\ r=\sqrt[]{\frac{A}{\pi}} \end{gathered}[/tex]Therefore, the approximate distance from the edge of the garden to the center (radius) is:
[tex]\begin{gathered} r=\sqrt[]{\frac{100}{3.14}} \\ r=\sqrt[]{31.85} \\ r\approx5.64ft \end{gathered}[/tex]That is the answer.
what is the common difference in the sequence 25,20,15,10...?
We have a arithmetic sequence: 25, 20, 15, 10...
Tipically, arithmetic sequences can be written in recursive form as:
[tex]a_n=a_{n-1}+d[/tex]where a(n) and a(n-1) are consecutive terms and d is the common difference.
In this case, we can see that each term decreases by 5 units.
Then, we can describe this sequence as:
[tex]a_n=a_{n-1}-5[/tex]which means that d = -5.
Answer: the common difference is d = -5.
1/2 + 1/5 = * Your answer
We applied the rules for adding fractions with different denominators. This is a way to achieve this. Graphically, we do the operations in this way:
Answer:
7/10
Step-by-step explanation:
1/2 + 1/5
We need to get a common denominator
1/2 * 5/5 = 5/10
1/5 * 2/2 = 2/10
We can add these together
5/10 + 2/10 = 7/10
Solve for n: 400(1.16)^n=35,120
The given equation is:
[tex]400\left(1.16\right)^n=35120[/tex]It is required to solve the equation for the value of n.
Divide both sides of the equation by 400:
[tex]\begin{gathered} \frac{400\left(1.16\right)^n}{400}=\frac{35120}{400} \\ \\ \Rightarrow\left(1.16\right)^n=\frac{439}{5} \end{gathered}[/tex]Take the logarithm of both sides of the equation:
[tex]\begin{gathered} \log(1.16)^n=\log\left(\frac{439}{5}\right) \\ \text{ Apply the power property of logarithms:} \\ \Rightarrow n\log(1.16)=\log\left(\frac{439}{5}\right) \end{gathered}[/tex]Divide both sides by log (1.16):
[tex]\begin{gathered} \frac{n\log(1.16)}{\log(1.16)}=\frac{\log\left(\frac{439}{5}\right)}{\log(1.16)} \\ \Rightarrow n=\frac{\operatorname{\log}(\frac{439}{5})}{\operatorname{\log}(1.16)}\approx30.151 \end{gathered}[/tex]The value of n is about 30.151.
to qualify for a police academy, candidates must score in the top 21% on a general abilities test. assume the test scores are normally distributed and the test has a mean of 200 and a standard deviation of 20. find the lowest possible score to qualify
The lowest value that is needed in order to qualify is given as 216.128
What is z score?The Z score is used to calculate how many standard deviations above or below the mean the raw score is. It comes from:
z = (raw score - mean) / standard deviation
Given; mean of 200 and a standard deviation of 20
P(z > c) = 21% = 0.21
1 - P(z < c) = 0.21
P(z < c) = 0.79
we are to find the critical value of z using excel function
=NORM.S.INV(1-0.21)
= 0.806421247
To get the lowest value we would have to put the values in the formula
0.8064 = (x - 200)/20
0.8064 * 20 = (x - 200)
16.128 = (x - 200
take like terms
x = 200 + 16.128
x = 216.128
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Leave K in fraction form or round to at least 3 decimal places. Round off your final answer to the nearest hundredth.
By definition, an equation of a Combined Variation has the following form:
[tex]z=k(\frac{x}{y})[/tex]Where "k" is the Constant of Variation.
In this case, you know that the resistance "R" of a wire varies directly as its length and inversely as the square of its diameter.
Then, let be "R" the resistance of the wire (in ohms), "l" its length of the wire (in feet), and "d" its diameter (in inches).
Therefore, you can set up that the equation has this form:
[tex]R=k(\frac{l}{d^2})[/tex]According to the information given in the exercise, when:
[tex]\begin{gathered} l=3300 \\ d=0.16 \end{gathered}[/tex]The resistance is:
[tex]R=10357[/tex]Then, you can substitute values into the equation and solve for "k":
[tex]\begin{gathered} 10357=k(\frac{3300}{(0.16)^2}) \\ \\ (10357)(\frac{(0.16)^2}{3300})=k \end{gathered}[/tex][tex]k\approx0.080[/tex]Therefore, you can set up the following equation that represents this situation (using the value of "k"):
[tex]R=0.080\cdot\frac{l}{d^2}[/tex]Hence, if:
[tex]\begin{gathered} l=2900 \\ d=0.15 \end{gathered}[/tex]You can substitute these values into the equation and then evaluate, in order to find the corresponding resistance. This is:
[tex]\begin{gathered} R=0.080\cdot\frac{(2900)}{(0.15)^2} \\ \\ R\approx10311.11 \end{gathered}[/tex]Therefore, the answer is:
[tex]10311.11\text{ }ohms[/tex]
what is the density of the oak board? show your work.
I think this is a physics problem.
I'll read it
a) A rectangular prism and a cylinder
b) Volume of the log = pi*r^2 x h
Volume of the log = 3.14*5^2* 30
Volume of the log = 2355 in^3
density = weight / volume
density = 4263 / 2355
density = 1.81 lb/in^3 This is the result
Please help me this question I couldn’t understand it please.
Given:
Length of a rectangle is a+1
width of a rectangle is a
[tex]\begin{gathered} \text{Perimeter}=2(a+1+a) \\ =2(2a+1) \\ =4a+2 \end{gathered}[/tex]Question 2 of 5
√16 =
Ο Α. 6
OB. 4
C. 8
OD. 3
Answer:
Uhm if this is the square root of 16 it's B.) 4
The graphs of y = f(x) and y = g(x) are shown on the coordinate plane below.y = g(x) 10y = f(x)210-10-9-8-7 -6 -5 -4 -3 -2 -1 0- 1-26789 10-6-107If g(x) = k·f(x), what is the value of k?
Answer:
Explanation:
The first step is to find the equation of both lines. The equation of a line in the slope intercept form is expressed as
y = mx + c
where
m is the slope
c is the y intercept. It is the value of y at the point where the line cuts the vertical axis.
The formula for calculating slope is expressed as
m = (y2 - y1)/(x2 - x1)
where
y1 and y2 are the y coordinates of selected initial and final points on the line.
x1 and x2 are the x coordinates of the selected initial and final points on the line.
Considering labelled points on y = f(x),
when x1 = 0. y1 = - 3
when x2 = 2, y2 = 1
m = (1 - - 3)/(2 - 0) = 4/2 = 2
y intercept, c = - 3
The equation would be
y = f(x) = 2x - 3
Considering labelled points on y = g(x),
when x1 = 0. y1 = 6
when x2 = 2, y2 = - 2
m = (- 2 - 6)/(2 - 0) = - 8/2 = - 4
y intercept, c = 6
The equation would be
y = g(x) = - 4x + 6
The solution of both equations is the coordinate of the point of intersection. Thus,
Solution = (1.5, 0)
Given that
g(x) = kf(x), it means that
- 4x + 6 = k(2x - 3)
Substituting x = 1.5 into the equation, we have
- 4(1.5) + 6 = k(2 * 1.5) - 3
- 6 + 6 = 3k - 3
0 = 3k - 3
3k = 3
k = 3/3
k = 1
To solve for x, you divide each side by what number?(4.5)x = 264.5456
Answer
4.5
Step-by-step explanation
Given the equation:
[tex]4.5x=26[/tex]Dividing at both sides by 4.5, we get:
[tex]\begin{gathered} \frac{4.5x}{4.5}=\frac{26}{4.5} \\ x=\frac{52}{9} \end{gathered}[/tex]Answer:
Divide each side by 4.5
Step-by-step explanation:
(4.5)x = 264.5456
We want to isolate x
Divide each side by 4.5
(4.5)x / 4.5 = 264.5456/ 4.5
x =58.78791
I need help with this assignment!! I already did A and B! I need help with the rest.
Given:
The roller-coster is moving in the trajectory of this curve
[tex]f(x)=3x^4-18x^3-21x^2+144x-108[/tex]Step by step solution:
To solve this complete problem we need to draw the estimated graph of this function, so that we can answer this question easily.
First of all, we need to find the roots of the given equation,to plot the curve:
let us put the random numbers that may satisfy the equation:
Let us put x = 1:
[tex]\begin{gathered} f(x)=3x^4-18x^3-21x^2+144x-108 \\ \\ f(1)=3-18-21+144-108 \\ \\ f(1)=\text{ 0} \end{gathered}[/tex]From here we can say that 1 is the root of the equation.
We will now divide this function from (x-1), so that we can get the cubic equation:
We will use long division method for division, the result we get after the division is:
[tex]f(x)=(x-1)(3x^3-15x^2-36x+108)[/tex]We will now try to factorize the cubic equations, by putting the random numbers that may satisfy the equation:
let us put x = 2:
[tex]\begin{gathered} f(x)=(x-1)(3x^3-15x^2-36x+108) \\ \\ f(2)=(2-1)(3(2)^3-15(2)^2-36(2)+108) \\ \\ f(2)=(1)(24\text{ }-\text{ 60 - 72 +108}) \\ \\ f(2)=0 \end{gathered}[/tex]From here we can say that f(2) is also the root of this cubic
We will now divide the cubic equation with (x-2), so we can break the cubic into quadratic:
Upon division the cubic equation break into following factors:
[tex]\begin{gathered} =(x-2)(3x^2-9x-54) \\ \\ which\text{ further simplified into:} \\ \\ =(x-2)(x-6)(x+3) \end{gathered}[/tex]From here we have found out four roots of the initial function that are:
x = 1,2,6,-3
Now we can easily plot the curve:
This is estimated curve, there are no sharp edges.
On the basis of this curve, we can easily answer all the questions related to this curve.
reshma is making a necklace using green beads and purple beads in a ratio represented on the following double number line fill in the missing values on the diagram and then answer the following question
From the double number line, we can see that the corresponding number of Green and Purple beads needed in each case are stated.
[tex]\begin{gathered} 4\text{ Gre}en\text{ }\rightarrow5\text{ Purple} \\ 8\text{ Gr}een\text{ }\rightarrow10\text{ Purple} \\ \cdot \\ \cdot \\ 20\text{ Gre}en\text{ }\rightarrow\text{ 25 Purple} \end{gathered}[/tex]Therefore, for 20 Green beads she will need to use 25 Purple Beads.
[tex]25\text{ Purple Beads}[/tex]234Weight (pounds)OA. The weight of the package is a function of the number ofpackages.B. The shipping cost is a function of the number of packages.C. The shipping cost is a function of the weight of the package.OD. The weight of the package is a function of the shipping cost.SUBMIT
The graph representing the relation between the shipping cost and the weight of packages is shown.
It is required to choose which function is true about the functional relationship.
Since the shipping cost is at the vertical axis and the weight of packages is at the horizontal axis, it follows that the shipping cost is a function of the weight of the package.
The answer is C.
The function in the form f(x)=2/5x-5 is graphed below. What is the value of x when f(x)=-9?
help me; its all explained in the picture thank you
The mean, mode, and mid-range of the given numbers are 13,13,16 respectively.
What are the mean, mode, and range?The total of all the numbers is represented by the mean. The median is the number in the center of an ordered list. The most frequent number is the mode. The highest number less the smallest number is the range.
Mean = sum of the number/ total no. of observations
Mean = 117/ 9
Mean = 13
Mode: The unique number that repeatedly comes
Given,
9, 9, 10, 11, 13, 13, 13, 14, 25
Mode = 13
Range: Deduct the smaller number from the greater one.
Range = 25-9
Range = 16
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200×200 dividend by 20
We need to find the value of 200×200 dividend by 20
So,
[tex]\frac{200\cdot200}{20}=\frac{40000}{20}=2000[/tex]Divide 30.4cm into 8 equal parts.
Find the length of each part.
Answer:
304/10÷8/1
304/10×1/8
38/10
3.8
or
304÷8
=38
I could use some help on math I’m really struggling
We need to find how much will be left after 6 half-lives of a radioactive isotope starting with 130g.
One way to write the amount N of radioactive isotope left after a time t, with an initial amount N₀ and a half-life τ is:
[tex]N=N_0\left(\frac{1}{2}\right)^{t\text{ /}\tau}[/tex]Notice that when t = τ, we have:
[tex]N=\frac{N_0}{2}[/tex]In this problem, we have:
[tex]\begin{gathered} N_0=130g \\ \\ t=6\tau \end{gathered}[/tex]Then, we obtain:
[tex]N=130g\left(\frac{1}{2}\right)^{6\tau\text{ /}\tau}=130g\left(\frac{1}{2}\right)^6=\frac{130g}{64}\cong2\text{ g}[/tex]Therefore, rounding to the nearest gram, the answer is 2 grams.
Pls help now You play a game that requires rolling a 6 sided die then randomly choosing a colored card from a deck containing 5 red cards,4blue cards, and 8 yellow cards find the probability that you will roll 3 on the die and choose a yellow card
Find the probability that you will get a 3 on a roll of a die. Since there is only one 3 in a die and there are 6 sides in a die, divide 1 by 6.
[tex]P(3)=\frac{1}{3}[/tex]Find the probability that you will get a yellow card. Divide the number of yellow cards by the total number of cards.
[tex]\begin{gathered} P(y)=\frac{8}{5+4+8} \\ =\frac{8}{17} \end{gathered}[/tex]Since the two events are independent, multiply the obtained probabilities.
[tex]undefined[/tex]The length of a rectangle is 5 inches more than the width. The perimeter is 42 inches. Find the length and the width of the rectangle.The width of the rectangle is ___ cubit inches, square inches or inches ? and the length of the rectangle is ____ cubit inches, square inches or inches?
Given
perimeter = 42 inches
length of a rectangle is 5 inches more than the width.
Find
width, length
Explanation
Let width of rectangle = x inches
length = 5 + x
Perimeter of rectangle = 2 (l + b) = 2(5+x+x) = 42
[tex]\begin{gathered} 2\times(5+x+x)=42 \\ 5+2x=21 \\ 2x=16 \\ x=8 \end{gathered}[/tex]width = 8 inches
Length = 5 + 8 = 13 inches
Final Answer
The width of the rectangle is 8 inches.
The length of the rectangle is 13 inches.
hello) i need some help with b) include an explanation if not a problem, thanks in advance)
What would happen if X doesn't lie on OM? If it is true that X lies on OM, if we suppose the opposite then we should have a contradiction, so the only way the contradiction doesn't happen is that it is true
Statements1. We know that BX:XA = 1:2
2. We know that M is the middle point between B and P
We need to prove that X lies on OM
Let's suppose X doesn't lie on OM
By 2, we know that 2BM = BP
If X doesn't lie on OM then the intersection between OM and BA is not X
Let's say the line that goes from O to the line BP and intersects BA on X is OX', where X'≠M
y varies directly as x. y =84 when x=6. Find y when x=12y= ?
If y varies directly as x, we have that
[tex]y\propto x[/tex]Then
[tex]y=kx(where\text{ k is a constant\rparen}[/tex][tex]\begin{gathered} When\text{ y=84 , x= 6} \\ y=kx \\ 84=6k \\ k=\frac{84}{6}=14 \end{gathered}[/tex]The relationship between x and y is given as
[tex]\begin{gathered} y=kx \\ y=14x \end{gathered}[/tex]Therefore when x= 12, y=?
[tex]\begin{gathered} y=14x \\ y=14\times12=168 \end{gathered}[/tex]Hence, the value of y when x = 12 is 168
Final answer: y = 168
Marco and Jazmin each bought trees to plant from Lowe’s. Marco spent $188 on 7 lemon trees and 9 orange trees. Jazmin spent $236 on 13 lemon trees and 9 orange trees. How much did lemon trees cost? How much did orange trees cost?
Let x be the cost of each Lemon tree and y the cost of eache orange tree. So we get that
[tex]\begin{cases}7x+9y=188 \\ 13x+9y=236\end{cases}\rightarrow6x=48\rightarrow x=\frac{48}{6}=8[/tex]having that the lemon tree cost $8 we get that
[tex]56+9y=188\rightarrow9y=188-56=132\rightarrow y=\frac{132}{9}=\frac{44}{3}[/tex]so each orange tree cost $44/3