The front of an a frame cabin in a national park is the shape of a triangle, with an area of 189 ft.². If the height is 1 foot less than twice the base, find the base and the height of the front of the cabin.
Answers
it can't be a negative value, since it's a measurement unit, thus it can't be -27.
so, anyhow, base is "b", and the height is "2b - 1".
Final answer:
The student needs to solve a quadratic equation to find the base and height of a triangle using the area formula and the given relationship between height and base. The solution involves substitution, expansion, and application of the quadratic formula or factoring.
Explanation:
The problem involves finding the base and height of a triangular front of an A-frame cabin based on its given area and a relationship between the height and base. It's a typical quadratic equation problem found in the high school mathematics curriculum when dealing with geometry and algebra.
To find the base (b) and height (h) of the triangle, we first use the area formula of a triangle A = 1/2 × base × height. We know that the area (A) is 189 ft² and that the height (h) is 1 foot less than twice the base, so h = 2b - 1. Substituting h into the area formula, we get 189 = 1/2 × b × (2b - 1). Solving this quadratic equation, we find the values for the base (b) and substitute back to find the height (h).
The process entails expanding the equation, moving all terms to one side to set the equation to zero, and then using the quadratic formula or factoring to find the value of b. Once the base is found, we use the relationship h = 2b - 1 to determine the height.
Related Questions
How many outfits are possible from 4 pairs of jeans 6 shirts and 2 pairs of shoes? Assume that outfit consists of 1 pair of jeans, 1 shirt, and 1 pair of shoes
Answers
and the rest would a least be missing either pants or shoes
multiply jeans by shirts by shoes
so 4 x 6 x 2 = 48 different combinations
What is the median of the data set given below? 19, 22, 46, 24, 37, 16, 19, 33
Answers
16,19,19,22,24,33,37,46....the median is the middle number. If there is an even number of numbers, like this one, there will be 2 middle numbers. Find those 2 middle numbers, divide them by 2, and that is ur median.
start from the ends and count inwards till u come to the 2 middle numbers...
(22 + 24) / 2 = 46/2 = 23
** now if there would have been an odd number of numbers, u would just have 1 middle number, and that number would be the median.
16,19,19, 22, 24, 33, 37, 46
If the total amount of data [let's take it as n] is odd, then the median is data on [[tex] \frac{n}{2} [/tex]].
But here the total amount of data is even [n=8], so the median is the average of the data on [[tex] \frac{n}{2} [/tex]] and [[tex] \frac{n}{2} +1[/tex]].
So the median is
[tex] \frac{22+24}{2} = \frac{46}{2} = 23[/tex]
Determine the order in which an inorder traversal visits the vertices of the given ordered rooted tree.
Answers
Explanation:
The left subtree of the root approaches first which is namely the tree rooted at b. There again the left subtree approaches first so the list begins with d. Afterwards that approaches b the root of this subtree and then the right subtree of b specifically in order f, e and g. At that point approaches the root of the whole tree and finally its right child. Thus the answer is d,b,f,e,g,a,c.
What key features of a polynomial can be found using the fundamental theorem of algebra and the factor theorem?
Answers
Answer:
The key fundamental theorem of algebra says that degree of polynomial is equal to number of zeros in a function.The Factor Theorem states that a first degree binomial is a factor of a polynomial function if the remainder, when the polynomial is divided by the binomial, is zero.The probability that you will win a game is 0.18. if you play the game 504 times, what is the most likely number of wins?
Answers
A store is offering a 20% discount on all sales over $50 if you purchase a T-shirt and a pair of jeans for $62.50 what is the amount of the discount you would receive
Answers
($62.50*0.2) = 12.5
Therefore, you would get a discount of $12.50. Total price would be $62.50 - $12.50.
The discount on the purchase of a T-shirt and a pair of jeans costing $62.50 with a 20% discount policy is $12.50.
A store's discount policy offers 20% off for purchases over $50. The question involves calculating the discount amount when a T-shirt and a pair of jeans are purchased together for $62.50. To determine the discount, we simply multiply the total purchase amount by the discount rate.
Step-by-Step Calculation:
First, confirm the purchase amount qualifies for the discount. The combined cost of the T-shirt and jeans is $62.50, which is above $50, so the purchase qualifies for the discount.
Calculate the discount by multiplying the total purchase amount by the discount rate: $62.50 × 0.20 (which is the same as 20%).
Discount Amount = $62.50 × 0.20 = $12.50.
Therefore, the amount of the discount the customer would receive on the purchase of the T-shirt and jeans is $12.50.
If, on average, Bob can make a sale to every 3rd person that comes into his store, how many people must come into Bob�s store if he wanted to make approximately fifteen (15) sales?
Answers
n = 3 * 15 = 45
What is the simplified form of 5 - 4y + 2x - 3y - 2 + 5x?
A. 3 - 7y + 7x
B. 3 + 7y + 7x
C. 7 + 7y + 7x
D. 3 - y + 7x
show steps pls!
Answers
5-2=3
-4y-3y=-7y
2x+5x=7x
Put these combined values together:
3-7y+7x
Final answer: A
M(5, 7) is the midpoint of rs The coordinates of S are (6, 9). What are the coordinates of R?
(5.5, 8)
(7, 11)
(10, 14)
(4, 5)
Answers
[tex]x_M = \frac{x_R + x_S}{2}[/tex]
[tex]5 = \frac{x + 6}{2}[/tex]
[tex]10 = x + 6[/tex]
[tex]x = 4[/tex]
[tex]y_M = \frac{y_R + y_S}{2}[/tex]
[tex]7 = \frac{y + 9}{2}[/tex]
[tex]14 = y + 9[/tex]
[tex]y = 5[/tex]
[tex]\therefore R(4, 5)[/tex]
What is the positive root of the equation x 2 + 5x = 150?
Answers
Then factor to (x+15)(x-10)=0
The roots are -15 and 10
The positive root is 10
The positive root of the given quadratic function is x = 10.
How to solve the quadratic function?We have the equation:
[tex]x^2 + 5x = 150[/tex]
First, we rewrite it as:
[tex]x^2 + 5x - 150 = 0[/tex]
Using the Bhaskara's formula, we will get:
[tex]x = \frac{-5 \pm \sqrt{5^2 - 4*(-150)*1} }{2} \\\\x = \frac{-5 \pm 25 }{2}[/tex]
The positive solution is:
x = (-5 + 25)/2 = 10
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Find the percent equivalent to the ratio 8 to 20
Answers
12 to 25 ==> 12/25 = 0.48 ===> 0.48*100 = 48%
divide the ratio:
8/20 = 0.4
0.4 = 40%
Guess the value of the limit (correct to six decimal places). (if an answer does not exist, enter dne.) lim hâ0 (4 + h)5 â 1024 h
Answers
Recall the definition of the derivative of a function [tex]f(x)[/tex] at a point [tex]x=c[/tex]:
[tex]f'(c)=\displaystyle\lim_{h\to 0}\frac{f(c+h)-f(c)}h[/tex]
We can then see that [tex]f(c)=c^5[/tex], and by the power rule we have [tex]f'(c)=5c^4[/tex]. Then replacing [tex]c=4[/tex], we arrive at
[tex]\displaystyle\lim_{h\to0}\frac{(4+h)^5-4^5}h=5\times4^4=1280[/tex]
Alternatively, we could have expanded the binomial, giving
[tex]\dfrac{(4+h)^5-4^5}h=\dfrac{(4^5+5\times4^4h+10\times4^3h^2+10\times4^2h^3+5\times4h^4+h^5)-4^5}h[/tex]
[tex]=\dfrac{1280h+640h^2+160h^3+20h^4+h^5}h[/tex]
[tex]=1280+640h+160h^2+20h^3+h^4[/tex]
and so as [tex]h\to0[/tex] we're left with 1280, as expected.
Based on the table, which statement best describes a prediction for the end behavior of the graph of f(x)? As x → ∞, f(x) → –∞, and as x → –∞, f(x) → ∞ As x → ∞, f(x) → ∞, and as x → –∞, f(x) → ∞ As x → ∞, f(x) → ∞, and as x → –∞, f(x) → –∞ As x → ∞, f(x) → –∞, and as x → –∞, f(x) → –∞
Answers
Answer: The correct option is B, i.e., "As x → ∞, f(x) → ∞, and as x → –∞, f(x) → ∞".
Explanation:
From the table it is noticed that the first row represents the value of x and the second row represents the value of f(x).
The value of f(x) is 14 at x = -5, after that the value of f(x) is decreased as the value of x increases.
The value of f(x) remains unchanged when the value of x approaches to 0 from 1.
The value of f(x) is -6 at x = 0, after that the value of f(x) is increased as the value of x increases.
From the table it is noticed that as the value of x approaches to positive infinity the value of f(x) is also approaches to positive infinity.
[tex]f(x)\rightarrow\infty \text{ as }x\rightarrow\infty[/tex]
From the table it is noticed that as the value of x approaches negative infinity the value of f(x) is also approaches to positive infinity.
[tex]f(x)\rightarrow\infty \text{ as }x\rightarrow-\infty[/tex]
These statement are shown in second option, therefore the second option is correct.
Answer:
The end behavior of the graph of the function f(x) is:
f(x) → ∞, and as x → –∞, f(x) → ∞ As x → ∞,
Step-by-step explanation:
Based on the table we could observe that the function f(x) is increasing to the left of -1 as well to the right of -1 and it attains the minimum value to be -6.
Hence, it can be predicted that the end behavior of the graph of the function f(x) goes to infinity in the left and also it goes to infinity in the right.
Hence, the statement that best describes the end behavior of the graph of f(x) is:
f(x) → ∞, and as x → –∞, f(x) → ∞ As x → ∞.
Which of the following are vertical asymptotes of the function y=3cot(1/2x)-4? A. 3pi B. 2pi C. pi/2 D. 0
Answers
cot is reciprocal of tangent.
cot = 1/tan, Find where tan = 0, since 1/0 goes to infinity.
tan(a) = 0 when a = 0,pi
Now Apply coefficient inside cot function.
x/2 = 0,pi
x = 0, 2pi
Therefore, vertical asymptotes occur when x = 0 or 2pi
Answer:
B and D
Step-by-step explanation:
A total of 279 tickets were sold for the school play. they were either adult or student tickets. the number of student tickets sold was two times the number of adult tickets sold. how many adult tickets were sold?
Answers
a+s=279, using a from above in this equation gives you:
a+2a=279 combine like terms on left side
3a=279 divide both sides by 3
a=93
So 93 adult tickets were sold.
Final answer:
The number of adult tickets sold for the school play is 93, calculated by setting up an equation based on the information that the total tickets sold were 279 and student tickets were two times the number of adult tickets.
Explanation:
The question asks us to determine the number of adult tickets sold when a total of 279 tickets were sold for the school play, and the number of student tickets is two times the number of adult tickets sold. We can set up an equation to solve this problem.
Let A be the number of adult tickets, and S be the number of student tickets. The problem gives us two equations:
S = 2A (The number of student tickets is two times the number of adult tickets)
A + S = 279 (The total number of tickets sold is 279)
Substituting the first equation into the second gives us:
A + 2A = 279
3A = 279
A = 279 / 3
A = 93
Therefore, 93 adult tickets were sold for the school play.
Find the length of the third Angelou a triangle given that the first two angels are 35 and 70 show your work
Answers
angles in a triangle = 180 degrees
70 +35 = 105 degrees
180-105 = 75 degrees
3rd angle = 75 degrees
a total of 321 tickets were sold for the school play. They were either adult tickets or student tickets. The number of student tickets sold was two times the number of adult tickets sold. How many adult tickets were sold
Answers
total tickets= ST + AT
subbing in ST=2*AT into total=ST+AT
Total= 2*AT + AT
Total=3AT
321=3AT
AT=321/3=107
Total= ST +AT
321= ST + 107
321-107=ST
ST=214
The adult child radio at a local daycare center is 3 to 16.at the same rare how many adults are needed for 48 children?
Answers
To find the number of adults needed for 48 children at a daycare with an adult to child ratio of 3:16, we set up a proportion and solve for the unknown number of adults. We cross-multiply and divide to find that 9 adults are required.
The question asks us to calculate how many adults are needed for 48 children in a daycare center given an adult to child ratio of 3:16. To solve this, we need to set up a proportion based on the ratio and solve for the number of adults needed.
We have the ratio of adults to children as 3:16, which means for every 3 adults, there are 16 children. To find out how many adults are needed for 48 children, we set up the proportion:
3 adults / 16 children = x adults / 48 children
Now we cross-multiply and solve for x:
(3 adults) / (48 children) = (16 children)
144 = 16x
x = 144 / 16
x = 9
Therefore, 9 adults are needed to take care of 48 children at the daycare center.
An ellipse has vertices along the major axis at (0, 8) and (0, –2). The foci of the ellipse are located at (0, 7) and (0, –1). What are the values of a, b, h, and k, given the equation below? (y-k)^2/a^2+(x+h)^2/b^2=1
Answers
now hmm, from the provided vertices and focus point, you can pretty much see what "a" is, half of the major axis, is just 5.
now, the center is from either vertex to half-way up, or "a" units up, so say from -2 + 5, is at 3, so the center is at 0, 3.
now, the distance from a focus point to the center, is 4 units, like say from 0, 3 up to 0,7.
[tex]\bf \textit{ellipse, vertical major axis}\\\\ \cfrac{(y-{{ h}})^2}{{{ a}}^2}+\cfrac{(x-{{ k}})^2}{{{ b}}^2}=1 \qquad \begin{cases} center\ ({{ h}},{{ k}})\\ vertices\ ({{ h}}, {{ k}}\pm a)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad \sqrt{{{ a }}^2-{{ b }}^2}\\ ----------\\ h=0\\ k=3\\ a=5\\c=4 \end{cases} \\\\\\ \cfrac{(y-3)^2}{5^2}+\cfrac{(x-0)^2}{b^2}=1[/tex]
now, let' s find "b".
[tex]\bf c=\sqrt{a^2-b^2}\implies c^2=a^2-b^2\implies b^2=a^2-c^2 \\\\\\ b=\sqrt{a^2-c^2}\implies b=\sqrt{5^2-4^2}\implies b=3[/tex]
so, just plug that in.
Answer: The values for a, b, h, and k are a = 5, b = 3, h = 0, k = -3.
Step-by-step explanation: In this problem, we know ellipse has vertices along the major axis at (0, 8) and (0, -2). The foci of the ellipse are located at (0, 7) and (0, -1). We are asked to determine the values of a, b, h, and k.
We were also then provided with the equation for vertical eclipse:
[tex]\frac{(x-h)^2}{b^2} + \frac{(y -k)^2}{a^2}[/tex]
Before we begin, we need to first define our values for a, b, h, and k.
a - distance to vertices from the centerb - distance to co-vertices from the center(h, k) - represents the center of the eclipseThe first step, we need to determine the center of the eclipse. We can use the midpoint formula to determine the midpoint between the vertices along the major axis: (0, 8) and (0, -2).
[tex]M = (\frac{x_{1} +x_{2} }{2} , \frac{y_{1} + y_{2} }{2} )[/tex]
[tex]M = (\frac{0 + 0}{2} , \frac{-2 + 8}{2} )\\M = (0, 3)[/tex]
We now know that our center (h, k) is (0, 3). Which means our values for h and k are 0 and 3. Next, we have to determine our values for a and b. Considering the center of our eclipse is not at the center, we can use one of our vertices to determine our value for a.
[tex]V_{1}[/tex] = (h, k±a)
(0, 8) = (0, 3±a)
3 ± a = 8
±a = 5
Now, we know that a = 5. For us to get b, we need to use this formula: [tex]c^2 = a^2 - b^2[/tex]. Let's rewrite this formula, so we can focus on getting our b-value.
[tex]c^2 - a^2 = -b^2[/tex]
For us to use this formula, we need to determine our c value. To find our c-value, we have use of our foci points: (h, k±c). C is the units away/further from the center towards our foci points.
(0, 3±c) = (0, 7)
3 + c = 7
7 - 3 = c
4 = c
Now, we know that our value for c is 4. Now, let's plug into the formula.
[tex](4)^2 - (5)^2 = -b^2\\16 - 25 = -b^2\\\frac{-9}{-1} = \frac{-b^2}{-1} \\\sqrt{b^2} = \sqrt{9} \\b = 3[/tex]
Our value for b is 3. If we put into our eclipse formula:
[tex]\frac{(x-0)^2}{3^2} + \frac{(y -(-3))^2}{5^2}[/tex]
A telephone pole cast a shadow that is 34 m long find the height of the telephone pole if a statue that is 36 cm tall cast a shadow 77 cm long ?
Answers
solve for "p".
Answer:
16 cm approximately
Step-by-step explanation:
We are given that a telephone pole cast shadow that is 34 m long .We are given that a statue that is 36 cm long and shadow of statue is 77 cm long.
We have to find the length of telephone pole
Let height of pole
Using direct proportion
[tex]\frac{x}{34}=\frac{36}{77}[/tex]
By multiply property of equality then we get
[tex]x=\frac{36}{77}\times 34[/tex]
x=[tex]\frac{1224}{77}[/tex]
x=15.896 cm
Hence, the height of telephone pole=15.896 cm=16 cm approximately
4. Meagan invests $1,200 each year in an IRA for 12 years in an account that earned 5%
compounded annually. At the end of 12 years, she stopped making payments to the
account, but continued to invest her accumulated amount at 5% compounded annually for
the next 11 years.
a. What was the value of the Ira at the end of 12 years?
b. What was the value of the investment at the end of the next 11 years?
c. How much interest did she earn?
Answers
[tex]\bf \qquad \qquad \textit{Future Value of an ordinary annuity}\\ \left. \qquad \qquad \right.(\textit{payments at the end of the period}) \\\\ A=pymnt\left[ \cfrac{\left( 1+\frac{r}{n} \right)^{nt}-1}{\frac{r}{n}} \right][/tex]
[tex]\bf \qquad \begin{cases} A= \begin{array}{llll} \textit{accumulated amount}\\ \end{array} \begin{array}{llll} \end{array}\\ pymnt=\textit{periodic payments}\to &1200\\ r=rate\to 5\%\to \frac{5}{100}\to &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\to &1\\ t=years\to &12 \end{cases}[/tex]
[tex]\bf A=1200\left[ \cfrac{\left( 1+\frac{0.05}{1} \right)^{1\cdot 12}-1}{\frac{0.05}{1}} \right]\implies A\approx 19100.55[/tex]
part B)
so, for the next 11 years, she didn't make any deposits on it and simple let it sit and collect interest, compounded annually at 5%.
[tex]\bf \qquad \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \ \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\to &\$19100.55\\ r=rate\to 5\%\to \frac{5}{100}\to &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\to &1\\ t=years\to &11 \end{cases} \\\\\\ A=19100.55\left(1+\frac{0.05}{1}\right)^{1\cdot 11}\implies A\approx 32668.42[/tex]
part C)
well, for 12 years she deposited 1200 bucks, that means 12 * 1200, or 14,400.
now, here she is, 12+11, or 23 years later, and she's got 32,668.42 bucks?
all that came out of her pocket was 14,400, so 32,668.42 - 14,400, is how much she earned in interest.
a. The value of the Ira at the end of 12 years is $19,100.55.
b. The value of the investment at the end of the next 11 years is $32,668.43.
c. Interest earn is $18,288.43.
a. Using this formula to determine the value of the Ira at the end of 12 years
A=Pmt [(1+r)^n-1]/r
Let plug in the formula
A=1,200[(1+0.05)^12-1]/0.05
A=1,200[(1.05)^12-1]/0.05
A=$1,200(0.795856)/0.05
A=$955.02759/0.05
A=$19,100.55
b. The value of the Ira at the end of 11 years is:
$19,100.55(1+0.05)^11
=$19,100.55(1.05)^11
=$19,100.55(1.710339358)
=$32,668.43
c. Interest earn
Total investment=1200(12)
Total investment= $14,400
Now let calculate the interest earned
Interest earned = $32,668.43 - $14,400
Interest earned= $18,288.43
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What is a non example of a rate
Answers
What's 10⁄12 written as a fraction in simplest form?
A. 5⁄6
B. 10⁄6
C. 3⁄5
D. 5⁄12
Answers
A. 5/6
How many different committees can be formed from 10 teachers and 41 students if the committee consists of 2 teachers and 2 students?
Answers
where C(n,r) represents the number of ways (order not important) we can choose r objects out of n, then
Number of ways to choose teachers = 10 choose 2 = C(10,2), and
number of ways to choose students = 41 choose 2 = C(41,2)
So the number of different committees
= C(10,2)*C(41,2)
= 45*820
= 36900
Bob drove from home to work at 50 mph. After work the traffic was heavier, and he drove home at 30 mph. His driving time to and from work was 1 hour and 4 minutes. How far does he live from his job?
Answers
The formula we need to solve this problem is:
Distance = Speed * Time
2.
Bob traveled the distance from work to his house at 30 mph, for 1 hour 4 minutes.
let's convert 1 hour 4 minutes to hours.
60 minutes are 1 hour
so 4 minutes are (4 minutes *1 hour)/ (60 minutes)= 0.067 hour.
Thus 1 hour 4 minutes = 1.067 h
The distance is Speed*Time= 30 mph*1.067 h=32.01 m
Answer: 32.01 miles
The blades of a windmill turn on an axis that is 40 feet from the ground. The blades are 15 feet long and complete 3 rotations every minute. Write a sine model, y = asin(bt) + k, for the height (in feet) of the end of one blade as a function of time t (in seconds). Assume the blade is pointing to the right when t = 0 and that the windmill turns counterclockwise at a constant rate.
a is the .
The vertical shift, k, is the... length of a blade, height of a windmill, or numbers of rotations per minute.
a =
k =
Answers
The rotational speed is
ω = (3 rev/min)*(1 min/60 s)*(2π rad/rev) = π/10 rad/s
The period is
T = 2π/ω = 20 s
The amplitude of the rotational motion is equal to the radius of 15 ft.
At t=0, the height of the tip of the blade is horizontal to the right, therefore its motion relative to the ground is given by
[tex]y=15 sin( \frac{ \pi t}{10})+40 [/tex]
A plot of the motion is shown below.
20
5
30
Step-by-step explanation:
Line segment AB is congruent to line segment CD.
A.AB overbar similar to CD overbar
B.AB overbar congruent to CD overbar
C. AB overbar equal to CD overbar
D. AB overbar element to CD overbar
Answers
When two line segments are congruent, what it really means is that the length of the line segments are equal. Congruent is usually interchanged with the word equal since it means the same: equal lengths. However, congruent is very specific in the sense that it means “equal lengths”. When we say equal alone, it may refer to different properties, congruent however is very specific.
Now the answer to this question should be:
B. AB overbar congruent to CD overbar
Not similar, not equal, and definitely not element. What we mean in this case is “equal lengths”, therefore the perfect word is congruent.
Margaret plans to deposit $500 on the first day of each of the next five years, beginning today. if she earns 4% compounded annually, how much will she have at the end of five years?
Answers
Deposits: = $ $3,424.81
Interest Earned: $424.81
The question is unclear, I calculated $500 yearly deposit with the compound.
Which number is greater? 0 -1 -25 -50?
Answers
Remember that with positive numbers, 50 is greater than 1, but with negative numbers, -1 is greater than -50.
If that confuses you, think of it like this.
If you owe someone 1 dollar, you yourself have more than if you owe someone 50 dollars. You yourself have even more if you don't owe any money.
write an expression for the area of the rectangle