A trapezoid has an area of 20 cm2 and a height z cm. The lengths of the parallel sides are (2z + 3) cm and (6z – 1) cm. Find the height, z, of the trapezoid. In your final answer, include all of the formulas and calculations necessary.
Answers
[tex]\bf A=\cfrac{h(a+b)}{2}\quad \begin{cases} A=20\\ a=6z-1\\ b=2z+3\\ h=z \end{cases}\implies 20=\cfrac{z[(6z-1)~+~(2z+3)]}{2} \\\\\\ 20=\cfrac{z(8z+2)}{2}\implies 20=\cfrac{2z(4z+1)}{2}\implies 20=z(4z+1) \\\\\\ 20=4z^2+z\implies 0=4z^2+z-20[/tex]
[tex]\bf \qquad \qquad \textit{quadratic formula}\\\\ \begin{array}{llccll} 0=&{{ 4}}z^2&{{ +1}}x&{{ -20}}\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array} \qquad \qquad z= \cfrac{ - {{ b}} \pm \sqrt { {{ b}}^2 -4{{ a}}{{ c}}}}{2{{ a}}} \\\\\\ z=\cfrac{-1\pm\sqrt{1+320}}{8}\implies z=\cfrac{-1\pm\sqrt{321}}{8}\implies z\approx \begin{cases} \boxed{2.1146}\\ -2.3646 \end{cases}[/tex]
since the height is just a length unit, it can't be -2.3646.
Answer:
2.11 cm
Step-by-step explanation:
As you can see in the picture I put the sides in the trapezoid. Now, let's use the area's formula:
A = [tex]\frac{(B+b)*h}{2}[/tex]
where B is the Largest parallel side, b the smallest parallel side and h is the height.
Now,
20 = [tex]\frac{(6z-1+2z+3)*z}{2}[/tex]
20 = [tex]\frac{(8z^{2}+2z}{2}[/tex]
20 = [tex]4z^{2}+z[/tex]
[tex]4z^{2}+z-20=0[/tex]
Then, we use the cuadratic formula z= [tex]\frac{-b+-\sqrt{b^{2}-4ac} }{2a}[/tex] where a=4, b=1 and c=-20.
z= [tex]\frac{-1+-\sqrt{1-4(4)(-20)} }{8}[/tex]
z = [tex]\frac{-1+-\sqrt{1+320} }{8}[/tex]
z = [tex]\frac{-1+-\sqrt{1+320} }{8}[/tex]
z = [tex]\frac{-1+-17.91}{8}[/tex]
As we are searching lenght we choose the positive z value
z = [tex]\frac{-1+17.91}{8}[/tex]
z= [tex]\frac{16.91}{8}[/tex]
z = 2.11 cm.
Related Questions
Peter has only quarters and dimes in his coin collection. If he has three times as many quarters as dimes, which is an expression for the number of quarters he has in terms of the number of dimes?
Answers
The expression for the number of quarters Peter has in terms of the number of dimes is [tex]\( \frac{3}{4}d \)[/tex].
Let's denote the number of dimes Peter has as [tex]\( d \)[/tex]. According to the problem, Peter has three times as many quarters as dimes. Therefore, if we let [tex]\( q \)[/tex] represent the number of quarters, we can write the relationship between the number of quarters and dimes as:
[tex]\[ q = 3d \][/tex]
However, the question asks for an expression that gives the number of quarters in terms of the number of dimes, but using the same variable [tex]\( d \)[/tex] to represent the total value of the dimes in dollars. Since each dime is worth $0.10, the total value of the dimes in dollars is:
[tex]\[ \text{Total value of dimes} = 0.10d \][/tex]
To find the number of quarters in terms of the total value of the dimes in dollars, we need to divide the total value of the dimes by the value of one quarter, which is $0.25, and then multiply by 3 because there are three times as many quarters as dimes:
[tex]\[ q = \frac{0.10d}{0.25} \times 3 \][/tex]
Simplifying the fraction [tex]\( \frac{0.10}{0.25} \)[/tex] gives us [tex]\( \frac{1}{2.5} \)[/tex], which simplifies further to [tex]\( \frac{2}{5} \)[/tex]. Therefore:
[tex]\[ q = \frac{2}{5}d \times 3 \][/tex]
[tex]\[ q = \frac{3}{5} \times 2d \][/tex]
[tex]\[ q = \frac{3}{5} \times 2 \times \frac{d}{1} \][/tex]
[tex]\[ q = \frac{3}{5} \times \frac{2d}{1} \][/tex]
[tex]\[ q = \frac{3}{5} \times d \times 2 \][/tex]
[tex]\[ q = \frac{3}{5} \times d \times \frac{2}{1} \][/tex]
[tex]\[ q = \frac{3}{5} \times \frac{2d}{1} \][/tex]
[tex]\[ q = \frac{3 \times 2d}{5} \][/tex]
[tex]\[ q = \frac{6d}{5} \][/tex]
However, we must remember that the original relationship was [tex]\( q = 3d \)[/tex], not . This means we made a mistake in our calculation. Let's correct it:
[tex]\[ q = 3d \][/tex]
Since each quarter is worth $0.25, the total value of the quarters in dollars is:
[tex]\[ \text{Total value of quarters} = 0.25q \][/tex]
[tex]\[ \text{Total value of quarters} = 0.25 \times 3d \][/tex]
[tex]\[ \text{Total value of quarters} = 0.75d \][/tex]
Now, to express the number of quarters [tex]\( q \)[/tex] in terms of the total value of the dimes in dollars using the same variable [tex]\( d \)[/tex], we need to adjust our equation to account for the value difference between dimes and quarters. Since the value of the dimes is given in dollars as [tex]\( d \)[/tex], and each quarter is worth $0.25, we can express the number of quarters as:
[tex]\[ q = \frac{d}{0.25} \times 3 \][/tex]
[tex]\[ q = \frac{d}{\frac{1}{4}} \times 3 \][/tex]
[tex]\[ q = d \times 4 \times 3 \][/tex]
[tex]\[ q = 4d \times 3 \][/tex]
[tex]\[ q = 12d \][/tex]
But this is not the expression we are looking for, as it gives us the number of quarters in terms of the total value of the dimes in dollars, not in terms of the number of dimes. We need to divide by 10 to convert the total value of the dimes in dollars back to the number of dimes:
[tex]\[ q = \frac{12d}{10} \][/tex]
[tex]\[ q = \frac{3}{4}d \times 4 \][/tex]
[tex]\[ q = 3d \][/tex]
This is the correct expression, as it gives us the number of quarters in terms of the number of dimes, with [tex]\( d \)[/tex] representing the number of dimes, not their value in dollars.
5+10x+5.15=60.15 what's the value of x?
Answers
Add the common like terms: 5 + 5.15 = 10.15
New equation:
10.15 + 10x = 60.15
Subtract the 10.15 from 60.15
10.15 + 10x = 60.15
- 10.15 -10.15
________________
10x = 50
Now divide 10x by 10 and 50 by 10.
[tex] \frac{10x}{10} [/tex] = [tex] \frac{50}{10} [/tex]
x = 5
5 is your final answer.
I hope this helped!!
find a linear function h such that h(3)=7 and h(-1)= 14. what is h (1/2)?
Answers
To find the slope (m), the equation is (y2-y1) / (x2-x1). In this case, x1 = 3, y1 = 7, x2 = -1, and y2 = 14.
So, we plug in these values into the equation to get (14 - 7)/(-1 - 3) = 7/-4 = -1.75. m = -1.75
Now to find the b value in the equation y = mx + b, we use the values we know and plug them into the equation. So,
7 = -1.75(3) + b
7 = -5.25 + b
b = 12.25
Check if this works with the other equation:
14 = -1.75(-1) + 12.25
= 1.75 + 12.25 = 14, checks out.
So, we know the function h is h(x) = -1.75 (x) + 12.25
We are asked to solve h(1/2), so simply plug it in for x. (I used 0.5 instead of 1/2 to make it simpler).
Doing this, we get
-1.75(0.5) + 12.25 = 11.375.
So, h (1/2) = 11.375
Hope this helps!
What is the area of a banner with 4 2/3 ft and 1 1/2 ft
Answers
4 2/3 x 1 1/5
4.666666666666667 x 1.5 = 7
The function d(x) =3x +2 determine how many donuts need to be purchased for a meeting, where x is the number of people at the meeting. The office manager uses h(d(x)) to find the amount of money to bring for the donuts purchase. The function h(x) = 2x + 7 . Solve for how much money to bring when there are 8 people In the meeting.
A.25
B.26
C.52
D.59
Answers
The number of sofas a factory produces varies directly with the number of hours the machinery is operational. Suppose the factory can produce 455 sofas in 48 hours. What is an equation that relates the number of sofas produced, n, with the amount of time, t, in hours? What is the graph of your equation?
Answers
455 = 48k
k = 455 / 48 = 9.479
Required equation is n = 9.479 t Answer
The graph will be a straight line passing through the origin with slope 9.479.
An automotive repair center charges $45 for any part of the first hour of labor, and $25 for any part of each additional hour. Which of the following is a correct cost?
A. C(t) = 145 for 5 < x ≤ 6
B. C(t) = 145 for 6 < x ≤ 7
C. C(t) = 170 for 5 < x ≤ 6
D. C(t) = 170 for 6 < x ≤ 7
Answers
C(t) = 45 + 25(h-1) where h is the number of hours worked.
We can check for each option in turn:
Option A:
Inequality 5 < x ≤ 6 means the hour is between 5 hours (not inclusive) to 6 hours (inclusive)
Let's take the number of hours = 5
C(5) = 45 + (5-1)×25 = 145
Let's take the number of hours = 6
Then substitute into C(6) = 45 + (6-1)×25 = 170
We can't take 145 because the value '5' was not inclusive.
Option B:
The inequality is 6 < x ≤ 7
We take number of hours = 6
C(6) = 25(6-1) + 45 = 170
We take number of hours = 7
Then C(7) = 25(7-1) + 45 = 195
Option C:
The inequality is 5 < x ≤ 6
Take the number of hours = 5
C(5) = 25(5-1) + 45 = 145
Take the number of hours = 6
C(6) = 25(6-1) + 45 = 170
We can't take the value 145 as '5' was not inclusive in the range, but we can take 170
Option D:
6 < x ≤ 7
25(6-1) + 45 < C(t) ≤ 25(7-1) + 45
170 < C(t) ≤ 195
Correct answer: C
Write an equation in point-slope form of the line that passes through the point (−8, −2) and has a slope of m=5
Answers
To write the equation in point-slope form, we can use the formula y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Explanation:To write an equation in point-slope form, we can use the formula y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. In this case, the given point is (-8, -2) and the slope is 5. Plugging in these values into the formula, we get:
y - (-2) = 5(x - (-8))
Simplifying the equation, we get:
y + 2 = 5(x + 8)
Therefore, the equation in point-slope form of the line that passes through the point (-8, -2) with a slope of 5 is y + 2 = 5(x + 8).
A batch of 140 semiconductor chips is inspected by choosing a sample of 5 chips. assume that 10 of the chips do not conform to customer requirements. the number of samples of 5 containing exactly one nonconforming chip is closest to:
Answers
p = 10/140 = 0.0714
The probability of having a good chip is
q = 1 - p = 0.9286
Out of a sample of 5, the probability of finding a defective chip is
₅C₁ p¹ q⁴ = 5*0.0714*0.9286⁴ = 0.2655
The number of defective chips ofu of 5 is 0.2655*5 = 1.3275 ≈ 1
Answer : 1 chip
I have a value that is 35% of the total value. I want to know what the total value is and what formula is used to work this out?
Answers
Add.
7 2/15 + 5 2/3 + 9 13/15
20 2/3
21 10/15
21 2/3
22 2/3
Answers
The only one that needs to be changed a little is 5 2/3 so that it will also have 15 as a denominator.
5 2/3 * 5/5 = 5 10/15
7 2/15 + 5 10/15 + 9 13/15
12 12/15 + 9 13/15
21 25/15
21 25/15
We can simplify this a bit further by dividing the fraction by 5.
21 25/15 = 21 5/3
21 5/3
Now change the improper fraction into a mixed fraction and add the whole number to the current whole number.
5/3 = 1 2/3
21 + 1 2/3 = 22 2/3
The answer is
D. 22 2/3
Adding together the whole number parts gives us 21. When we add the fractions, we get 25/15, which simplifies to 1 10/15. Adding this to our whole number sum gives us 22 10/15, or 22 2/3.
Explanation:To find the sum of these mixed numbers, you'll want to first add the whole number parts, and then add the fractions. In this case, adding together the whole numbers 7, 5, and 9 gives us 21.
The fractions 2/15, 2/3, and 13/15 can be added together by finding a common denominator. The common denominator for 15 and 3 is 15, therefore 2/3 becomes 10/15 when it is converted.
When we add 2/15, 10/15, and 13/15, we get 25/15, which can be reduced to 1 10/15.
When we add this to our whole number sum, we get 22 10/15,
which simplifies to 22 2/3.
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The function f(x) = 68(1.3)x represents the possible squirrel population in a park x years from now. Each year, the expected number of squirrels is ____ the number the year before.
Answers
This is the rate of change,
Answer:
1.3 times
Step-by-step explanation:
A car manufacturer wants to change the content of a certain automobile to have less steel in order to get better gas mileage. One hundred standard cars and 100 cars of the new content are built and test driven across country to determine the overall gas mileage. This study is an
Answers
The car manufacturer wants to test some changes that he wants to implement in his cars, which is why he has two control groups (standard cars and cars with the new content) to test those changes out. This is a typical experiment and not an observation as he is not really observing much.
Answer:
experiment
Step-by-step explanation:
Drag the blue labels onto the table to identify the data that is relevant to each hypothesis. then use the pink labels to indicate whether each hypothesis is supported or not supported by the data answers
Answers
Hypothesis 1 supported (potentially adapted wings), 2 needs more data, 3 and 4 not supported (scavengers and traffic constant).
Here's how to proceed:
1. Match Data to Hypotheses:
Hypothesis 1:
Relevant Data: "The wing shapes of swallows killed on roads differ from those of the general population."
Hypothesis 2:
Relevant Data: Not directly provided in the given options. More information is needed about the actual population size of cliff swallows living near roads over time.
Hypothesis 3:
Relevant Data: "Avian scavengers did not increase during this time, and terrestrial scavengers probably did not increase."
Hypothesis 4:
Relevant Data: "Car traffic stayed the same or increased during this time."
2. Evaluate Hypothesis Support:
Hypothesis 1:
Supported (additional data may be needed). The wing shape difference suggests potential adaptation, but more evidence would strengthen the conclusion.
Hypothesis 2:
Cannot be determined without direct data on population size changes.
Hypothesis 3:
Not supported (probably not a factor). The data indicates scavenger populations haven't increased, suggesting they aren't significantly affecting the number of road-killed swallows found.
Hypothesis 4:
Not supported (probably not a factor). The data shows car traffic has remained steady or increased, making it less likely to be the cause of a decrease in observed road-killed swallows.
Complete Question:
What is 48,371 rounded to the nearest thousand?
Answers
what is the exponent used in cubing a number
Answers
So, the answer is 3.
The average weight of a mature human brain is approximately 1400 grams. What is the equivalent weight in pounds? Use the conversion equivalency 1 kilograms (kg) = 1000 grams (g) and 2.20 pounds (lb) = 1 kg.
Answers
Find the value of y log4 64=y
Answers
Answer:
y=log4 64=2.6665.
Step-by-step explanation:
We are given that logarithmic expression
y=log 464
By using logarithmic rules
Substitute the decimal point after end digit and then put zero after decimal point
We can write as
y=log464.0
To put the decimal point after one digit from left then we move two steps.Therefore ,we write 2 on left side of the decimal point in final result
Now, we see the value of 46 at 4 from log table then we get the value of 46 at 4 is 6665
Therefore , y=log464=2.6665
Hence, the value of y=2.6665
The value of y to the equation log₄(64) = y is y = 3.
The given logarithmic equation is :
y = log₄(64)
This can be written in exponential form as :
4^(y) = 64
It is known that :
4 × 4 × 4 = 64
So,
4³ = 64
Hence the value of y = 3.
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Use the result from part c to find the two solutions to the equation 2x2−3x−5=0. enter the two solutions separated by a comma. (the order is not important.)
Answers
The quadratic formula can be used to find the solutions of a quadratic equation. In this case, the equation is 2x² - 3x - 5 = 0. Using the quadratic formula, the two solutions are x = 2 and x = -1.
Explanation:To find the solutions to the equation 2x² - 3x - 5 = 0, we can use the quadratic formula. The formula states that the solutions of any quadratic equation ax² + bx + c = 0 can be calculated using the formula:
x = (-b ± √(b² - 4ac)) / 2a
In this case, a = 2, b = -3, and c = -5. Substituting these values into the formula, we get:
x = (-(-3) ± √((-3)² - 4(2)(-5))) / (2(2))
Simplifying further, we have:
x = (3 ± √(9 + 40)) / 4
x = (3 ± √49) / 4
x = (3 ± 7) / 4
Therefore, the two solutions to the equation 2x² - 3x - 5 = 0 are x = (3 + 7) / 4 = 2 and x = (3 - 7) / 4 = -1.
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What is the relationship between the 3s in the number 24,335
Answers
I hope this helped
Find f(5) for f (x)=1/4 (2)^x
Answers
f(5)=1/4 (2)^5 Apply the exponent to the value inside the parentheses
f(5)=1/4 x 32 Multiply 1/4 and 32
f(5)= 8
Your answer is A. 8
The correct option is A. 8
Given f (x)=1/4 (2)^x.
[tex]f(x)[/tex] [tex]=\frac{1}{4} 2^{x}[/tex]
We have to calculate F(5), means put x = 5.
So, [tex]f(5)=\frac{1}{4} 2^{5}[/tex]
[tex]f(5)=\frac{1}{4} \times 32[/tex]
[tex]f(5)=8[/tex]
Hence [tex]f(5)=8[/tex] .
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Lines are drawn through the point (2, 8) and the points given below. Select two points that correspond with lines with negative slopes.
(-2, -6)
(0, 9) (
1, -3)
(5, 6)
(10, 11)
Answers
To find two points with negative slopes, we can calculate the slopes of the lines passing through the point (2, 8) and each of the given points. The pairs of points with negative slopes are (-2, -6) and (1, -3).
Explanation:A line has a negative slope when it goes down from left to right. To find two points with negative slopes, we can calculate the slopes of the lines passing through the point (2, 8) and each of the given points.
Using the slope formula, slope = ∆y / ∆x, we can calculate the slopes:
-2, -6: slope = (-6 - 8) / (-2 - 2) = -14 / -4 = 3.5
1, -3: slope = (-3 - 8) / (1 - 2) = -11 / -1 = 11
Therefore, the pairs of points with negative slopes are (-2, -6) and (1, -3).
Is 3 a good estimate for 3.4x0.09
Answers
Chelsea has four hours of free time on Saturday. She would like to spend no more than 2/3 of an hour on each activity. How many activities can she do during her time?
Answers
[tex]\bf \cfrac{\stackrel{\textit{total hours}}{4}}{\stackrel{\textit{hours per activity}}{\frac{2}{3}}}\implies \cfrac{\quad \frac{4}{1}\quad }{\frac{2}{3}}\implies \cfrac{4}{1}\cdot \cfrac{3}{2}\implies \cfrac{12}{2}\implies 6[/tex]
How can knowing the multiplication properties help you multiply 5x(2x6)?
Answers
What Is 22/55 in lowest terms
Answers
Hope this helps!
2/5
Time measured in minutes is an example of a(n) _______________ scale.
Answers
The answer can be written in statement form as,"Time measured in minutes is an example of a(n) interval scale."
What is an interval scale?A numerical scale known as an interval scale is one in which both the order of the variables and their differences are known. Using the Interval scale, variables with recognizable, consistent, and calculable differences are categorized.
It is easy to remember the primary role of this scale too, ‘Interval’ indicates ‘distance between two entities’, which is what Interval scale helps in achieving.
The value of time is given in minutes.
The given quantity can be measured on interval scale.
Hence, the required scale is known as interval scale.
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Do I add the lenghs of the sides of a figure and multiply them by the height to get the area?
Answers
The most basic shape, a square, or even a rectangle's area is found by multiplying the two different side lengths together. (length and width)
A triangle is the base * height / 2.
Trapezoid base 1 + base 2 *height / 2 or something like that.
Circle is [tex] \pi [/tex] r squared.
Just search it up for each one.
837,164 and 4,508 the value of 8
Answers
In 4,508, the value of 8 is the ones place
hope this helps
p=21+2w (solve for w)
Answers
plz make me the brainliest