The hypotenuse of a right triangle is 8 feet less than three times the shorter leg and the longer leg is 8 feet more than twice the shorter leg. find the lengths of the three sides of the triangle.
The sides of the right angle triangle are 20, 48, and 52.
We have given that,
The hypotenuse of a right triangle is 8 feet less than three times the shorter leg and the longer leg is 8 feet more than twice the shorter leg.
Suppose the shortest leg is x, the hypotenuse is then 3x-8, and the longer leg is 2x+8
We use the Pythagorean theorem
What is the Pythagorean theorem?[tex]hypotenous^2=side^2+side^2[/tex]
Therefore by using Pythagorean theorem
[tex]x^2+(2x+8)^2=(3x-8)^2\\x^2+(4x^2+32x+64)=(9x^2-48x+64)\\4x^2-80x=0[/tex]
x=0 or x=20
So the sides are 20, 48, and 52.
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Which equation represents a line with a greater slope and lesser y intercept than the line shown?
To start the school year, a teacher spends $54.35 in school supplies for her classroom. During the course of the year, she will restock her supplies and will spend $22 each time she restocks the supplies. The expression below describes this situation.
22x + 54.35
In this expression, what is the coefficient and what does it describe?
A.) x, which describes the number of times she restocks her supplies
B.) 22, which describes the amount she spends each time restocking her supplies
C.) 22x, which describes the total amount she spends each time she goes to the store
D.) 54.35, which describes the amount she originally spends on supplies
Which graph represents the system of inequalities?
y+32x>−3
y−5≤−6x
Rewrite the systems of inequalities in slope-intercept form, and then graph the lines using the respective inequalities. The shaded area that overlaps will represent the solution to these systems of inequalities.
Explanation:In order to determine a graph that represents the system of inequalities, y+32x > -3 and y-5 ≤ -6x, you will need to write the systems of inequalities in mathematical terms, known as slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Firstly, let's rewrite the first inequality as y > -32x - 3. This inequality suggests that y is greater than -32x - 3, therefore the area above the line will be shaded.
Next, the second inequality is rewritten as y ≤ 6x + 5. This suggests y is less than or equal to 6x + 5, thus the area below the line will be shaded.
Remember, when creating the graph, a solid line should be used for ≤ symbol and a dashed line for the > symbol. The overlapping region of the two inequalities on the graph is the solution to the system of inequalities.
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Y = 3x + 5 step 1 of 3: determine the slope and y-intercept of the equation above
John runs 4 miles in 30 minutes. at the same rate, how many miles would he run in 48 minutes?
33.33333333333333 in a fraction
dj14ven
What is the perimeter of the rectangle shown on the coordinate plane, to the nearest tenth of a unit?
12.7 units
16.9 units
24.0 units
33.9 units
The answer came from Rgwoot Ambitious
from another post in case anyone else needs it
What we know:
shape is rectangle which means the 2 long sides have equal distance and the 2 short sides have equal distance
we just need to find the distance of one long side and one short side for the perimeter which is the outline of the rectangle. Imagine the perimeter is the fence around the rectangle that you would probably have to paint every 3 years and the area would be where the grass would grow in the rectangle which you would probably have to cut every weekend.
perimeter=2l+2w
What we need to find: PERIMETER
Using pythagorean method a² +b²=h² to find length:
From point (-6,1) to point (3,8) is a rise of 9 and a run of 9 right to get from one point to another, those are my a and b in the pythagorean formula.
a² +b²=h²
(9)²+(9)²=h² substitution
81+81=h² simplified
162=h²
√162=√h2 used radical properties
√162=h length =√162
Using pythagorean method a² +b²=h² to find width:
From points (-6,-1) to point (-3,-4) is a down 3 units and left 3 units to reach from one point to another, these are my a and b for the pythagorean formula.
a² +b²=h²
(3)²+(3)²=h²
9+9=h²
18=h²
√18=√h²
√18=h this is the width=√18
Now we find perimeter:
p=2l+2w
p=2(√162)+2(√18)
p≈33.9
D. 33.9 units
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True or false: given a sample mean of 2.1 and a sample standard deviation of 0.7 from a sample of 10 data points, a 90% confidence interval will have a width of 2.36.
A 90% confidence interval will have a width of 2.36 which is false.
What is a confidence interval?Let μ be the mean, σ be the standard deviation, n be the ample size, and z be the z-score. Then we have
Confidence interval = μ ± z (σ / √n)
A sample mean of 2.1 and a sample standard deviation of 0.7 from a sample of 10 data points.
We know that the z-score for 90% of the confidence interval is 1.645. Then we have
Confidence interval = μ ± z (σ / √n)
Confidence interval = 2.1 ± 1.645 (0.7 / √10)
Confidence interval = 2.1 ± 0.364
For positive sign, then the width will be
⇒ 2.1 + 0.364
⇒ 2.46
A 90% confidence interval will have a width of 2.36 which is false.
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PRE CAL
f(x)=x^3+x^2+4x+4
Solve with Synthetic division and find all zeros of the function
Evaluate y= 50 X (0.3)^x for x= 2
A: 4.5
B: 750
C: 45
D: 225
The biggest problem when using metals (or cowry shells) for trading is what ?
write a percent between 25percent and 50 percent then write it as a decimal and as a fraction in simplest form
HELP PLEASE. A function f (x) is graphed on the coordinate plane.
What is the function rule in slope-intercept form?
The function rule in slope-intercept form is y = 3x + 9.
Explanation:The function rule in slope-intercept form is given by the equation y = mx + b, where m represents the slope of the line and b represents the y-intercept.
In this case, the given y-intercept is 9, so b = 9.
The slope of the line is 3, which means that for every increase of 1 on the horizontal axis (x-axis), there is a rise of 3 on the vertical axis (y-axis). Therefore, m = 3.
Putting these values into the slope-intercept form, we get the function rule to be y = 3x + 9.
what is the volume of a cube with 1/2 inch sides
Answer:
the answer is 1/8
Step-by-step explanation:
Icket sales tickets for a high school play are $3.00 each for students and $4.00 each for all others. find the total money collected from ticket sales if 315 student tickets are sold out of a total of 518 tickets
315 x 3 = 945
518 - 315 = 203
203 x 4 = 812
945 + 812 =1757
So the answer is $1,757 total
The total money collected from ticket sales for the high school play, with 315 student tickets sold at $3.00 each and 203 other tickets sold at $4.00 each, is $1,757.00.
To find the total money collected from ticket sales, we need to calculate the revenue from both student tickets and tickets sold to others. We know that 315 student tickets were sold at a price of $3.00 each, and the remaining tickets sold to others were at a price of $4.00 each. Since a total of 518 tickets were sold, the tickets sold to others amount to 518 - 315 = 203 tickets.
Revenue from student tickets = 315 tickets imes $3.00 per ticket = $945.00
Revenue from other tickets = 203 tickets imes $4.00 per ticket = $812.00
Total revenue from ticket sales = Revenue from student tickets + Revenue from other tickets = $945.00 + $812.00 = $1,757.00
Therefore, the total money collected from ticket sales is $1,757.00.
Assembly Line A produces 45 units in the same time that it takes Assembly Line B to produce 37 units. If Line B produces 555 units, how many units does Line A produce during the same time?
To determine the number of units produced by Assembly Line A when Line B produces 555 units, we use the ratio of their production rates (45/37) to find that Assembly Line A would produce 675 units.
The question asks how many units Assembly Line A produces when Assembly Line B produces 555 units, given that Line A produces 45 units in the same time that it takes Line B to produce 37 units. To find the answer, we can use the concept of ratio and proportion.
Since Line A produces 45 units at the same time Line B produces 37 units, we have a ratio of 45/37. With Line B producing 555 units, we can set up a proportion to find how many units Line A produces:
45/37 = x/555
Multiplying both sides of the equation by 555, we get:
x = (45/37) * 555
Calculating the value of x gives us:
x = 675
Therefore, Assembly Line A produces 675 units in the same time that Assembly Line B produces 555 units.
Explain why this statement isn’t completely accurate.
Choose the point slope form of the equation below that represents the line passing through the point (-6,4) and (2,0)
Answer:
So, the point-slope form of the equation that represents the line passing through the points [tex]\((-6,4)\) and \((2,0)\)[/tex] is:
[tex]\[ y - 4 = -\frac{{1}}{{2}}(x + 6) \][/tex]
Explanation:
To find the equation of the line passing through the points [tex]\((-6,4)\) and \((2,0)\)[/tex] , we can use the point-slope form of the equation of a line:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Where [tex]\((x_1, y_1)\)[/tex] is a point on the line, and [tex]\(m\)[/tex] is the slope of the line.
First, let's find the slope [tex]\(m\):[/tex]
[tex]\[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \][/tex]
Using the coordinates of the given points:
[tex]\[ m = \frac{{0 - 4}}{{2 - (-6)}} \][/tex]
[tex]\[ m = \frac{{-4}}{{2 + 6}} \][/tex]
[tex]\[ m = \frac{{-4}}{{8}} \][/tex]
[tex]\[ m = -\frac{{1}}{{2}} \][/tex]
Now that we have the slope, let's choose one of the given points to plug into the point-slope form. We'll use [tex]\((-6,4)\):[/tex]
[tex]\[ y - 4 = -\frac{{1}}{{2}}(x - (-6)) \][/tex]
[tex]\[ y - 4 = -\frac{{1}}{{2}}(x + 6) \][/tex]
[tex]\[ y - 4 = -\frac{{1}}{{2}}x - 3 \][/tex]
Now, let's simplify and put the equation in slope-intercept form:
[tex]\[ y = -\frac{{1}}{{2}}x - 3 + 4 \][/tex]
[tex]\[ y = -\frac{{1}}{{2}}x + 1 \][/tex]
Which is larger 1 meter or 105 centimeters?
what would be the maximum value for f(x) = −5x2 + 9
What is the equation of a line perpendicular to the line y=(4/9)x-2 that passes through the point (4,3)?
a. y= -(4/9)x-2
b. y= (9/4)x+6
c. y= -(9/4)x+12
d. y= -(9/4)x-12
(pls explain solution!!)
To find the equation of a line perpendicular to y=(4/9)x-2 that passes through the point (4,3), we need to determine the slope of the original line and then find the negative reciprocal of that slope. The equation of the perpendicular line is y = -(9/4)x + 12.
To find the equation of a line perpendicular to y=(4/9)x-2 that passes through the point (4,3), we need to determine the slope of the original line and then find the negative reciprocal of that slope. The original line has a slope of 4/9, so the perpendicular line will have a slope of -9/4. Using the point-slope form of a line, we can plug in the slope and the given point to find the equation of the perpendicular line:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope. Plugging in the values, we get:
y - 3 = -(9/4)(x - 4)
Simplifying the equation, we get:
y = -(9/4)x + 12
Therefore, the equation of the line perpendicular to y=(4/9)x-2 that passes through the point (4,3) is y = -(9/4)x + 12.
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What is the equation of a line that passes through the point (4, 2) and is perpendicular to the line whose equation is y=x/3−1 ? Enter your answer in the box.
Lines can be parallel, perpendicular or have no relationship at all.
The equation of the line is: [tex]\mathbf{y = -3x + 14}[/tex]
First, we calculate the slope of: [tex]\mathbf{y = \frac x3 - 1}[/tex]
A linear equation is represented as:
[tex]\mathbf{y = mx + c}[/tex]
Where:
m represents the slope
So, by comparison:
[tex]\mathbf{m = \frac 13}[/tex]
From the question, we understand that the required line is perpendicular to [tex]\mathbf{y = \frac x3 - 1}[/tex]
This means that, the slope (m2) of the required line is:
[tex]\mathbf{m_2 = -\frac 1m}[/tex]
So, we have:
[tex]\mathbf{m_2 = -\frac 1{1/3}}[/tex]
[tex]\mathbf{m_2 = -3}[/tex]
The equation of the line is:
[tex]\mathbf{y = m_2(x - x_1) + y_1)}[/tex]
Where:
[tex]\mathbf{(x_1,y_1) = (4,2)}[/tex]
So, we have:
[tex]\mathbf{y = -3(x - 4) + 2}[/tex]
Open bracket
[tex]\mathbf{y = -3x + 12 + 2}[/tex]
[tex]\mathbf{y = -3x + 14}[/tex]
Hence, the equation of the line is: [tex]\mathbf{y = -3x + 14}[/tex]
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B is the midpoint of AC. If AB =x +5 and BC = 2x -11, find the measure of AB
What is the cosine of angle a?
a. 8/17
b. 15/8
c. 15/17
d. 8/15
After my salary was increased by 12%, it was $62,720$. What was my salary before the increase?
find the interval(s) over which the function is increasing
To find the intervals over which a function is increasing, you need to find its derivative, set it greater than 0 and solve for x. The solutions give the intervals where the function is increasing.
Explanation:In mathematics, to find the intervals over which a function is increasing, you first need to find its derivative. The derivative of a function at a certain point gives the slope of the tangent line at that point. If the derivative is positive, the function is increasing. If it's negative, the function is decreasing.
For example, let's say we have a function f(x) = x^3 - 3x. The derivative of this function is f'(x) = 3x^2 - 3. To find the interval where the function is increasing, we set the derivative greater than 0 and solve for x: 3x^2 - 3 > 0. The solution to this inequality gives the intervals over which the function is increasing .
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30 POINTS!!!!
Tom and Martin both worked 9 hours this week and sold $360 worth of meals. Their hourly wages are $2.15 and $2.50, respectively. In addition, Tom's percent of tips was 15% while Martin's was 20%. Who made the most money this week between Tom and Martin?
Given
Tom and Martin both worked 9 hours this week and sold $360 worth of meals
Tom's percent of tips was 15% while Martin's was 20%
Find out the most money this week between Tom and Martin
To proof
In this question take two cases
FIRST CASE
Martin worked 9 hours
sold worth of meals = $360
Martin's percent of tip = 20%
First convert 20% in decimal form
= [tex]\frac{20}{100}[/tex]
= 0.20
than the equation becomes
Martin makes money = 9 × 2.50 + 0.20 × 360
= 22.5 + 72
= $ 94.5
SECOND CASE
Tom worked 9 hours
Tom's percent of tips = 15%
sold worth of meals = $360
First 15 % is written in decimal form
= [tex]\frac{15}{100}[/tex]
= 0.15
than the equation becomes
Tom makes money = 9 × 2.15 + 0 .15× 360
= 19.35 + 54
= $ 73.35
Thus martin makes money is $94.5
Thus tom makes money is $73.35
Hence Martin makes more money than tom
Hence proved
Answer: Tom made more money than Martin
Step-by-step explanation:
Test results said this was the right answer
how do you solve 5x^2+30x=10
Erica is buying a $205,000 home. She has decided to purchase 2 points in order to lower her interest rate. The appraisal fee is $450, the processing fee is $575, and the title fee is $600. What is her total in closing costs?
Answer: $5,725
Step-by-step explanation:
Given: The cost of home = $205,000
If there is no down payment, then the mortgage amount = $205,000
We know that one point costs 1 % of mortgage amount .
2 points = 2 % of the mortgage amount =[tex]0.02\times205,000=\$4,100[/tex]
[tex]\text{Closing costs=}\text{cost of points + appraisal fee+processing fee+ title fee}[/tex]
[tex]=\$4,100+\$450+\$575+\$600=\$5,725[/tex]
Hence, her total in closing costs = $5,725.