Answer:
Given:
[tex]\textsf{Shorter leg}=(x+5)[/tex][tex]\textsf{Longer leg}=3(x+5)-1[/tex][tex]\textsf{Hypotenuse}=2(x+5)+13[/tex][tex]\sf A=2.5P\quad \textsf{(where A is area and P is perimeter)}[/tex][tex]\begin{aligned} \textsf{Perimeter} & =\textsf{shorter leg + longer leg + hypotenuse}\\ & = (x+5)+[3(x+5)-1]+[2(x+5)+13]\\ & = x+5+3x+15-1+2x+10+13\\ & = 6x+42\end{aligned}[/tex]
[tex]\begin{aligned} \textsf{Area} & =\dfrac12\textsf{(shorter leg)(longer leg)}\\ & =\dfrac12(x+5)[3(x+5)-1]}\\ & =\dfrac12(x+5)(3x+14)\\ & = \dfrac12(3x^2+29x+70)\\ & = \dfrac32x^2+\dfrac{29}{2}x+35 \end{aligned}[/tex]
[tex]\begin{aligned} \textsf{Area} & =2.5 \sf (Perimeter)\\ \implies \dfrac32x^2+\dfrac{29}{2}x+35 & =\dfrac52(6x+42)\\ 3x^2+29x+70 & =5(6x+42)\\ 3x^2+29x+70 & =30x+210\\ 3x^2 -x-140 & =0\\ 3x^2-21x+20x-140 & =0\\ 3x(x-7)+20(x-7) &=0\\ (x-7)(3x+20)& =0\end{aligned}[/tex]
Therefore:
[tex](x-7)=0 \implies x=7[/tex]
[tex](3x+20)=0 \implies x=-\dfrac{20}{3}[/tex]
[tex]x=-\dfrac{20}{3}[/tex] is not a viable solution as when inputting this into the formula for the shorter leg, it gives a negative value:
[tex]\textsf{Shorter leg}=-\dfrac{20}{3}+5=-\dfrac53[/tex]
As distance cannot be negative, this is not a viable solution.
When x = 7:
[tex]\textsf{Shorter leg}=7+5=12[/tex]
[tex]\textsf{Longer leg}=3(7+5)-1=35[/tex]
[tex]\textsf{Hypotenuse}=2(x+5)+13=37[/tex]
[tex]\textsf{Perimeter}=6(7)+42=84[/tex]
[tex]\textsf{Area} & =\dfrac32(7)^2+\dfrac{29}{2}(7)+35=210[/tex]
Therefore, there is one viable solution. This solution in the form (x, A) is (7, 210)
A pan from the oven is sitting out to cool to room temperature. If the temperature difference between the pan and room temperature is currently 197°C and is decreasing by 6% every minute, how much above room temperature will the pan be in 15 minutes? If necessary, round your answer to the nearest tenth.
The temperature difference between the pan and room temperature would be 77.9⁰C in the next 15 minutes,
How to solve an exponential function?Let y represent the temperature difference after x minutes.
The difference between the pan and room temperature is currently 197°C and is decreasing by 6% every minute, hence:
y = 197(100% - 6%)ˣ
y = 197(0.94)ˣ
The temperature in the next 15 minute is:
y = 197(0.94)¹⁵ = 77.9⁰C
The temperature difference between the pan and room temperature would be 77.9⁰C in the next 15 minutes.
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Find the volume of the prism.
7.5m
8m
4m
What is the slope of a line parallel to the line whose equation is
6x – 10y = -100. Fully simplify your answer.
Answer:
3/5
Step-by-step explanation:
6x - 10y = -100
Write this equation in slope-intercept form: y = mx +b
-10y = -6x - 100
Divide the entire equation by (-10)
[tex]\dfrac{-10}{-10}y=\dfrac{-6}{-10}x-\dfrac{100}{-10}\\\\y =\dfrac{3}{5}x+10[/tex]
Slope = 3/5
Parallel lines have same slope.
Slope of the parallle line = 3/5
The diagram shows a right angled triangle. What is the value of h.
Round to 1 decimal place
Answer:
h = 13.4cm
Step-by-step explanation:
BAC + ACB = 90°
BAC + 48° = 90°
BAC = 42°
cos(BAC) = AB/BC
cos(42°) = h/18
h = 13.377 ≈ 13.4cm
For a standard normal distribution, find the approximate value of p (negative 0.78 less-than-or-equal-to z less-than-or-equal-to 1.16). use the portion of the standard normal table below to help answer the question.
The approximate value of P(-0.78 ≤ Z ≤ 1.16) is obtained being 0.6593
How to get the z scores?If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z score.
If we have [tex]X \sim N(\mu, \sigma)[/tex]
(X is following normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex])
then it can be converted to standard normal distribution as
[tex]Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)[/tex]
(Know the fact that in continuous distribution, probability of a single point is 0, so we can write
[tex]P(Z \leq z) = P(Z < z) )[/tex]
Also, know that if we look for Z = z in z-tables, the p-value we get is
[tex]P(Z \leq z) = \rm p \: value[/tex]
For this case, we have to find:
[tex]P(-0.78\leq Z \leq 1.16)[/tex]
It can be rewritten as:
[tex]P(-0.78\leq Z \leq 1.16) = P(Z \leq 1.16) - P(Z < -0.78) \\P(-0.78\leq Z \leq 1.16) = P(Z \leq 1.16) - P(Z \leq -0.78)[/tex]
The p-values for Z = 1.16 and Z = -0.78 from the z-table is found as 0.8770 and 0.2177 respectively, and therefore, we get:
[tex]P(-0.78\leq Z \leq 1.16) = P(Z \leq 1.16) - P(Z \leq -0.78)\\P(-0.78\leq Z \leq 1.16) = 0.8770 - 0.2177 = 0.6593[/tex]
Thus, the approximate value of P(-0.78 ≤ Z ≤ 1.16) is obtained being 0.6593
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Answer:
B.) The answer is 66% if you convert it from decimals
Given SA 108 units squared SA=192 units squared V-1408 units squared Find Volume of the smaller figure
Line k is graphed at right.
Write an equation of a line parallel to K
Write an equation of a line perpendicular to K
Answer:
See below ↓↓
Step-by-step explanation:
Equation of line k
y = mx + bm = Δy/Δx (ratio of change in y to change in x) = 2/3b = 3 (as y-intercept is [0,3])⇒ y = 2/3x + 3a. parallel line
For parallel lines, slope (m) remains the same, but the y-intercept (b) changes⇒ y = 2/3x + 1b. perpendicular line
For perpendicular lines, the slope (m) is the negative reciprocal of the original line and the value of b stays the same [it can be any point as long as it intersects the line]⇒ y = -3/2x + 1How do i solve this pls help
Answer:
7.07106781
simplified
7.07
Step-by-step explanation:
The area of a storage shed floor is 20 ft2. The length is 3 more than twice the width. Find the dimensions of the storage shed floor
Answer:
2.5 ft wide8 ft longStep-by-step explanation:
The given relation between length and width can be used to write an expression for area. The equation setting that equal to the given area can be solved to find the shed dimensions.
__
Given relationLet x represent the width of the shed. Then the length is (2x+3), and the area is ...
A = LW
20 = (2x+3)(x) . . . . . area of the shed
__
SolutionCompleting the square gives ...
2x² +3x +1.125 = 21.125 . . . . . . add 2(9/16) to both sides
2(x +0.75)² = 21.125 . . . . . . . write as a square
x +0.75 = √10.5625 . . . . . divide by 2, take the square root
x = -0.75 +3.25 = 2.50 . . . . . subtract 0.75, keep the positive solution
The width of the shed is 2.5 feet; the length is 2(2.5)+3 = 8 feet.
If the length of a cube is l then find the area of cross section
Answer:
The cross section of a cube is square...
and we know that ,Area of square = side²
so, Area of cross section= l²
Hope it helps you
help with this questions please!
math
Step-by-step explanation:
[tex]area \: of \: rectangle = length \times \: width[/tex]
Given
[tex]l = 2w - 3[/tex]
using this
[tex]area = l \times w = (2w - 3)(w) = 27[/tex]
[tex]2 {w}^{2} - 3w = 27[/tex]
[tex]2 {w}^{2} - 3w - 27 = 0[/tex]
[tex]2 {w}^{2} - 9w + 6w - 27 = 0[/tex]
[tex]2w(w - 4.5) + 6(w - 4.5) = 0[/tex]
[tex](2w + 6)(w - 4.5) = 0[/tex]
[tex]w = - 3 \: \: or \: \: w= 4.5[/tex]
Since size cannot be negative therefore
[tex]w = 4.5cm[/tex]
[tex]l = 2w - 3 = 9 - 3 = 6[/tex]
Hence length of the rectangle=6m
width of the rectangle=4.5m
Answer:
Let length and breadth of rectangle be x metres and y metres respectively.
xy = 27 ---- eqn 1
x = 2y-3 ---- eqn 2
Sub eqn 2 into eqn 1:
(2y-3) × y = 27
2y² - 3y = 27
2y² - 3y - 27 = 0
(y+3)(2y-9) = 0
y+3=0 or 2y-9=0
y=-3 (rejected) 2y=9
y=4.5
Sub y=4.4 into eqn 2:
x = 2(4.5) - 3
= 6
Hence,
length of rectangle = x metres
= 6m
breadth of rectangle = y metres
= 4.5m
Region R is bounded by the curves y = √x, y = 1, and x = 4. A solid has base R, and cross sections perpendicular to the x-axis are squares. The volume of this solid is
A. 4/3
B. 8
C. 7/6
D. 15/2
The cross sections have side length equal to the vertical distance between y = √x and y = 1, or |√x - 1|. The two curves meet at the point (1, 1), and y = √x meets x = 4 at (4, 2), so we'll be integrating with respect to x on the interval [1, 4]. Over this interval, √x ≥ 1, so |√x - 1| = √x - 1.
A cross section of thickness ∆x has volume
(√x - 1)² ∆x = (x - 2√x + 1) ∆x
Then the volume of the solid is
[tex]\displaystyle \int_1^4 (x - 2\sqrt x + 1) \, dx = \boxed{\frac76}[/tex]
is 6 and 9 like terms or unlike terms
Answer:
i think they are unlike terms
Step-by-step explanation:
A couch is discounted by $299. If the original price is $1250 , estimate the sale by price by first rounding each number to the nearest hundred
Answer: $950
1250-299=951
round off to nearest hundred, $950
The estimated sale price of the couch after rounding each number to the nearest hundred is $901.
To estimate the sale price of the couch, we first need to round each number to the nearest hundred.
Original price: $1250
Rounded to the nearest hundred: $1200
Discount amount: $299
Now, let's calculate the estimated sale price by subtracting the discount from the rounded original price:
Estimated Sale Price = Rounded Original Price - Discount Amount
Estimated Sale Price = $1200 - $299
Estimated Sale Price = $901
The estimated sale price of the couch after rounding each number to the nearest hundred is $901.
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How do I solve 20x + 21??
Answer:
Step-by-step explanation:
20x+21=0
20x=-21
x=-1.05
Surface Area of Cylinders 4 cellus 6 = SA = 2tr2 + 2nrh 2πη2 + 2πχh (Use 3.14 for 7.) Resources 5 in. Find the surface area. Help square inches 30 in. Do NOT round your answer. B If
Answer:
1099in²
Step-by-step explanation:
SA = [tex] \sf A=2\pi rh+2 \pi r² [/tex]
⇒ SA=2πrh+2πr²
⇒ SA = 2((3.14)(5)(30)+(2)(3.14)(5²)
⇒ SA = 2((3.14)(5)(30)+(2)(3.14)(25)
⇒ SA = 1099
Surface Area = 1,099 in²
A dryer and washer cost $936 combined the washer cost $86 more than the dryer what is the cost of the dryer
What is the measure of the missing angle?
1. 7h + 2h - 5h
Α 4h
Β. 14h
Answer:
A 4h
Step-by-step explanation:
[tex]7 + 2 - 5 = 9 - 5 = 4[/tex]
guys plsshelp meee!!!!!
Answer:
It's (5,4)
Step-by-step explanation:
hope this helps :)
Nikki went to a concert that started at 2:30pm.it ended at 4:00pm. How long was the concert?
started= 2:30 PM
ended= 4:00 PM
to find:the duration of the concert.
solution;( just find the difference of finishing time and starting time)
= 1 hour 30 mins
so, the duration of the concert is 1 hour 30 mins.
0400-0230
= 1 hour 30 minutes
hope this helps :)
Can someone please help me factor this
Answer:
[tex]\huge\boxed{\bf\:1}[/tex]
Step-by-step explanation:
[tex]\frac{ x ^ { 2 } -4x+3 }{ x ^ { 2 } -7x+12 } \times \frac{ x ^ { 2 } +2x-24 }{ x ^ { 2 } +5x-6 } ^ { }[/tex]
Take [tex]\frac{ x ^ { 2 } -4x+3 }{ x ^ { 2 } -7x+12 }[/tex] & factorise it at first.
[tex]\frac{ x ^ { 2 } -4x+3 }{ x ^ { 2 } -7x+12 } \\= \frac{\left(x-3\right)\left(x-1\right)}{\left(x-4\right)\left(x-3\right)}\\= \frac{x-1}{x-4}[/tex]
Now factorise the next set : [tex]\frac{ x ^ { 2 } +2x-24 }{ x ^ { 2 } +5x-6 } ^ { }[/tex].
[tex]\frac{ x ^ { 2 } +2x-24 }{ x ^ { 2 } +5x-6 } ^ { }\\= \frac{\left(x-4\right)\left(x+6\right)}{\left(x-1\right)\left(x+6\right)}\\= \frac{x-4}{x-1}[/tex]
Now, multiply the two simplified results.
[tex]\frac{ x ^ { 2 } -4x+3 }{ x ^ { 2 } -7x+12 } \times \frac{ x ^ { 2 } +2x-24 }{ x ^ { 2 } +5x-6 } ^ { }\\= \frac{x-1}{x-4}\times \frac{x-4}{x-1} \\= \frac{\left(x-1\right)\left(x-4\right)}{\left(x-4\right)\left(x-1\right)} \\= \boxed{\bf\: 1}[/tex]
[tex]\rule{150pt}{2pt}[/tex]
The perimeter of a rectangle is 24. write the function that describes its area in terms of one of the sides. if one side is a, the formula will be S= ____
The area of the rectangle whose perimeter is 24 units and one side is of 'a' units, in terms of 'a' is given as: S = a(12-a) unit²
How to find the area of a rectangle?Suppose that the two adjacent sides of a rectangle be of 'a' units and 'b' unit lengths.
Then, we get the area of that rectangle as:
[tex]S = a \times b \: \rm unit^2[/tex]
For this case, we're specfied that:
One of the side of the considered rectangle is of 'a' units length.The perimeter of the considered rectangle = 24 unitsLet the other side (adjacent) be of 'b' units length
Then, as perimeter of a rectangle = 2(sum of lengths of one pair of adjacent sides of the rectangle)
Therefore, we get:
[tex]24 = 2(a+b)\\\text{Dividing both the sides by 2}\\12 = a + b\\b = 12 - a[/tex]
We expressed 'b' in terms of 'a' so that we can represent the area of the considered rectangle in terms of 'a' alone.
The area of the rectangle is:
[tex]S =a \times b = a \times (12 - a) \: \rm unit^2[/tex]
Thus, the area of the rectangle whose perimeter is 24 units and one side is of 'a' units, in terms of 'a' is given as: S = a(12-a) unit²
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Naomi bought stock in a company two years ago that was worth
x
x dollars. During the first year that she owned the stock, it increased by 23%. During the second year the value of the stock increased by 26%. Write an expression in terms of
x
x that represents the value of the stock after the two years have passed.
The expression that represents the value of the stock after two years have passed is 1.5498x
What is the expression that reperesnts the value of the stock after two years?
Percentage is the fraction of an amount expressed as a number out of hundred. The sign used to represent percentages is %.
The value of the stock after year 1 = worth of the stock when it was bought x (1 + percentage increase in year one)
1.23x
The value of the stock after year 2 = worth of the stock in year 1 x (1 + percentage increase in year two)
1.26 x 1.23x = 1.5498x
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I need help please....
Answer:
[tex]y \geqslant - x - 4[/tex]
Step-by-step explanation:
y≥-x-4 is the required equation
The length, width and height of one of the small cubes is 1/3m.
Find the volume of the figure.
Find all possible values of x. The triangles are NOT drawn to scale.
Answer:
0 < x < 28
Step-by-step explanation:
The triangle inequality theorem states that the sum of any two sides of the triangle should be greater than the third side.
Therefore,
0 < x < 28All the possible values of x are 26 < x < 28.
What is Triangle?A triangle is a two dimensional figure which consist of three vertices, three edges and three angles.
Sum of the interior angles of a triangle is 180 degrees.
So a triangle has three sides.
We have the triangle inequality theorem which states that,
The length of any side of the triangle is always less than the sum of the other two sides.
And the range of the possible measure of the third side is the sum and the difference of the other two sides.
So x < 27 + 1 = 28
and,
x > 27 - 1 = 26
So 26 < x < 28
Hence the measures of third side is 26 < x < 28.
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A parking garage is located in the downtown area of a city. The
table below shows the cost for parking in the garage for different
amounts of time.
Hours Parked
Cost of Parking
1
$8.80 1 1/2 $10.70
4
$20.20
5
$24
7 1/2
$33.50
10
$43
a) What equation represents the cost of parking in the garage,
y, for x hours?
b) Sketch a graph to represent the cost of parking over time.
Answer:
a) y = 3.80x +5.00
b) see attached
Step-by-step explanation:
A graph shows the given table values lie on a straight line.
__
a)Finding the slope of the line is made easier by an appropriate choice of a pair of table values:
m = (y2 -y1)/(x2 -x1)
m = (24 -20.20)/(5 -4) = 3.80/1 = 3.80 . . . . using (4, 20.20) and (5, 24)
The y-intercept can likewise be found with an appropriate choice of table values. Solving the slope-intercept equation for b, we get ...
y = mx +b
b = y -mx
b = 8.80 -3.80 × 1 = 5.00 . . . . using the first table value
An equation that represents the cost of parking could be ...
y = 3.80x +5.00
__
b)A graph of the table values and the equation is shown in the attachment.
9 A boy takes a penalty 40 times against a goalkeeper. He scores 16 times.
What is the relative frequency of him scoring a penalty? zu
(b) If the boy took 200 penalties how many times would you expect him to score?
Answer:
I think for A It is 3.
Step-by-step explanation:
A car rental company charges $28 per day for a rented car and $0.50 for every mile driven. A second car rental
company charges $20 per day and $0.75 for every mile driven. What is the number of miles at which both
companies charge the same amount for a one-day rental?
Answer:
56 miles
Step-by-step explanation:
According to the question, lets say the miles at x they charge the same rental
34 + 0.50x = 20 + 0.75x (Solve equation)
0.75x - 0.50x = 34 - 20
0.25x = 14
x = [tex]\frac{14}{0.25}[/tex]
x = 56
So at 56 miles, they charge the same rental.