Answers
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Let x be the original side length of the square. The original area is therefore going to be x*x = x^2. For now we don't know the area but once we figure out x, we can compute the area.
The original side length x is increased by 7 ft to jump up to (x+7) ft
At the same time, the other dimension is reduced to (x-2) ft, which is a decrease of 2 ft
The old square is x ft by x ft with area x*x = x^2
The new rectangle is (x+7) ft by (x-2) ft
Let's find the area of this rectangle
Have a look at the image attachment to see how to multiply out (x+7)*(x-2) using the box method
The terms in red are x^2, 7x, -2x and -14 which are found by multiplying the headers
x times x = x^2
x times 7 = 7x
-2 times x = -2x
-2 times 7 = -14
The like terms are 7x and -2x which combine to 5x
Overall the result of multiplying out (x+7)*(x-2) is x^2+5x-14
Note: we can get the same result if we use the FOIL or distribution rule. The box method is a handy way to visually keep track of all the terms.
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The area of the rectangle, in terms of x, is x^2+5x-14
We are told that the area is 90 square feet
So equate the two expressions and solve for x
x^2+5x-14 = 90
x^2+5x-14-90 = 90-90
x^2+5x-104 = 0
(x-8)(x+13) = 0 ... see note below
x-8 = 0 or x+13 = 0
x = 8 or x = -13
note: find two numbers that multiply to -104 and add to 5. Those numbers are -8 and 13 which helps us factor
We ignore x = -13 as a negative side length makes no sense
The only practical root is x = 8
The side length of the original square is 8 feet
The area is
x*x = 8*8 = 64 square feet
We can check the answer by adding 7 to 8 to get 15, and subtracting 2 from 8 to get 6. The 8 by 8 square turns into a 15 by 6 rectangle. Since 15*6 = 90, we have confirmed the right answer.
Related Questions
This set of points is on the graph of a function.
{(−3, 9), (−1, 1), (0, 0), (2, 4)}
Which points are on the graph of the inverse?
Select each correct answer.
(0, 0)
(−9, 3)
(4, 2)
(1, −1)
What is the inverse of f(x)=4x+3 ?
f^−1(x)=1/4x−3/4
f^−1(x)=−1/4x+3/4
f^−1(x)=4x−3
f^−1(x)=−4x−3
What is the inverse of the function?
g(x)=−4/3x+2
g^−1(x)=−4/3x−2
g^−1(x)=3/4x+2/3
g^−1(x)=−3/4x+3/2
g^−1(x)=4/3x−2
Which functions are invertible?
Select each correct answer. (has pictures)
A.
B.
C.
D.
What is the inverse of f(x)=x4+7 for x≥0 where function g is the inverse of function f?
g(x)=x+7√4 , x≥−7
g(x)=x√4+7 , x≥0
g(x)=x−7√4, x≥7
g(x)=x√4−7 , x≥0
Answers
(-3,9) is one point on the original function, so (9,-3) is on the inverse. We swap x and y. The point (9,-3) isn't listed so we move onto the next.
(-1,1) is on the original so (1,-1) is on the inverse. Again we swap x and y. Choice D matches with this. So D is one of the answers.
Choice A is also an answer since (0,0) swaps to (0,0)
Choice C is also an answer since (2,4) swaps to (4,2)
In summary, the answers are A, C, D
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Problem 2)
Replace f(x) with y. Swap x and y. Then solve for y
f(x) = 4x+3
y = 4x+3
x = 4y+3
x-3 = 4y+3-3
x-3 = 4y
4y = x-3
4y/4 = (x-3)/4
y = (x-3)/4
y = x/4-3/4
y = (1/4)x-3/4
which matches with choice A, so choice A is the answer
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Problem 3)
Replace g(x) with y. Swap x and y. Then solve for y
g(x) = (-4/3)x + 2
y = (-4/3)x + 2
x = (-4/3)y + 2
x-2 = (-4/3)y + 2-2
x-2 = (-4/3)y
(-3/4)(x-2) = (-3/4)(-4/3)y
(-3/4)(x-2) = y
y = (-3/4)(x-2)
y = (-3/4)x+(-3/4)(-2)
y = (-3/4)x+3/2 ... answer is choice C
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Problem 4)
The first two are invertible while the last two are not. The last two fail the horizontal line test so they don't have an inverse.
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Problem 5)
It's a bit tricky to determine what the answer choices are saying but I'm assuming they are referring to fourth roots. If so, then we have
f(x) = x^4+7
y = x^4+7
x = y^4+7
x-7 = y^4+7-7
x-7 = y^4
y^4 = x-7
FourthRoot(y^4) = FourthRoot(x-7)
y = FourthRoot(x-7)
which is written as [tex]\sqrt[4]{x-7}[/tex]
The points on the inverse function are (9, -3), (1, -1), (0, 0), and (4, 2). The inverse of f(x)=4x+3 is[tex]f^{-1}(x) = -3 + \frac{1}{4}x[/tex]. The inverse of g(x)=−4/3x+2 is [tex]g^{-1}(x) = \frac{3}{4}x + \frac{2}{3}[/tex] . The inverse of f(x)=x^4+7 for x≥0 is [tex]g(x) = \sqrt{\frac{x-7}{4}}[/tex], for x≥7.
Explanation:To find the points on the graph of the inverse of the function, simply switch the x and y coordinates. So, the points on the graph of the inverse function would be (9, -3), (1, -1), (0, 0), and (4, 2).
For the inverse of the function f(x)=4x+3, you rearrange the equation to solve for x. That gives [tex]f^{-1}(x) = -3 + \frac{1}{4}x[/tex]Therefore, the correct answer is [tex]f^{-1}(x) = \frac{1}{4}x - \frac{3}{4}[/tex]
The inverse of the function g(x)=−4/3x+2 could be found in the same way, which gives[tex]g^{-1}(x) = \frac{3}{4}x + \frac{2}{3}[/tex]
As for the question regarding invertible functions, without the images it is not possible to provide an answer.
The inverse of the function [tex]f(x)=x^4+7[/tex] for x≥0 is calculated by swapping x and y, and then solving for y. This yields: [tex]g(x) = \sqrt{\frac{x-7}{4}}[/tex], for x≥7.
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Solve the equation for z and approximate the value of z
0.5*e^4z =13
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Can someone please just explain what I need to do? I'm so confused
The length of a text messaging conversation is normally distributed with a mean of 10 minutes and a standard deviation of 4 minutes. What is the probability that a conversation lasts longer than 20 minutes?
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What is the value of the expression when a = 4, b = 3 ,and c = 10?
3a/4b−c
1
4
6
12
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A trail is 2940 miles long. it begins in city a and ends in city
b. manfred has hiked three sevenths 3 /7 of the trail before. how many miles has he hiked?
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Final answer:
Manfred has hiked 1260 miles of the trail, which is calculated by multiplying the total length of the trail (2940 miles) by the fraction of the trail he has completed (3/7).
Explanation:
To calculate the distance Manfred has hiked on the trail, we multiply the total length of the trail by the fraction of the trail he has completed. In this case, Manfred has hiked three sevenths (3/7) of the trail. The total distance of the trail is 2940 miles.
Distance Hiked = (Total Distance of Trail) × (Fraction of Trail Completed)
Distance Hiked = 2940 miles × (3/7)
Distance Hiked = 2940 miles × 0.4286
Distance Hiked = 1260 miles
Therefore, Manfred has hiked 1260 miles of the trail.
which statement is equivalent to the inequality 14x-14>7x+7
Answers
14x - 7x > 7 + 14
7x > 21
x > 21/7
x > 3
PLZ HELP
Mrs. Kuss received a $50 Itunes gift card for her birthday. She bought one album for $9.99 and would like to see how much money she has left to buy single songs. If each song costs $1.29, how many songs can Mrs. Kuss buy with her remaining money? Show your work.
Answers
50.00-9.99 = 40.01
40.01 / 1.29 = 31.0
Bella babysits for two families each weekend. One family pays her $20. The other family pays her $22. Which expression represents the amount of money Bella has been paid after w weekends?
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How to solve 5x-9y=7 and -3x+7y=-5 using any method
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The x value will give 0.5 and the y value will be equal to -0.5
I hope this helps.
YOU'RE WELCOME :D
A cube is packed with decorative pebbles. If the cube has a side length of 4 inches, and each pebble weighs on average 0.5 lb per cubic inch, what is the total weight of the pebbles in the cube?
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This will give 32 pounds
1. Can √72 be completely simplified if it is broken up as √72=√9x8, explain why or why not?
2. How do you determine the result of taking the square root of a variable raised to an even power? For example √x∧28?
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2. sqrt x = x^½. Rules of indices states for a power of a power, you times them.
so (x^28)^½ = x^14
if MN = PQ AND PQ = RS then MN = RS
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Which of the following list of ordered pairs is a function?!!!
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What is 75% of 400 write evaluate an expression to find the answer.then explain how touse the model to justify the answer?
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75 /100 x 400 =
75 x400/ 100=
300
I like to do it this way :
100% = 400
1%= 400/100
= 4
75%= 4 x75
= 300
Hope this helps :))
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Is 55.56 greater than 55.65? I'm in a hurry so answer quick
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The function f(x) is shown in the graph.
What is the equation for f(x)?
Enter your answer in the box.
f(x)=_______
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Read the following two statements. Then, if possible, use the Law of Detachment to draw a conclusion. If Robbie wants to save money to buy a car, he must get a part-time job. Robbie started a new job yesterday at a grocery store. Robbie wants to save money. Robbie wants to buy a car. not possible If Robbie wants to save money, he must get a part-time job.
Answers
Final answer:
Using the Law of Detachment, we can conclude that Robbie wants to save money because he started a new job and the first condition states that he needs a job to save for a car.
Explanation:
Using the Law of Detachment, we can draw a conclusion from the following two statements:
If Robbie wants to save money to buy a car, he must get a part-time job.Robbie started a new job yesterday at a grocery store.Since Robbie started a new job, which is assumed to be part-time, we can conclude that Robbie wants to save money, as per the first statement which sets up a condition that needs to be fulfilled in order to save money. The specific intention of buying a car is not directly established, but it's often implied that people save for larger investments or expenses. The Law of Detachment only allows us to conclude propositions that directly follow from the given condition. Therefore, the direct conclusion we can draw here, based on the given statements and the Law of Detachment, is that Robbie wants to save money.
Final answer:
According to the Law of Detachment, Robbie wants to buy a car since he wants to save money and he has started a new job.
Explanation:
The Law of Detachment states that if we have a conditional statement of the form "If p, then q", and we know that p is true, then we can conclude that q is also true. In this case, let's analyze the given statements:
1. If Robbie wants to save money to buy a car, he must get a part-time job.
2. Robbie started a new job yesterday at a grocery store.
From statement 1, we can see that Robbie wanting to save money is the condition (p), and getting a part-time job is the conclusion (q). Since statement 2 confirms that Robbie has started a new job, which satisfies the condition (p), we can use the Law of Detachment to conclude that Robbie wants to save money (q) and therefore Robbie wants to buy a car (q). So, the conclusion is: that Robbie wants to buy a car.
Classify the expression 7x3 + 2x2 − 9 as a monomial, binomial, trinomial or polynomial.
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A packet of biscuits has a mass of 1 1/3 kg. Jason eats 2/5 of the packet. What is the mass of biscuits left?
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2 / 5 = 6 / 15;
So. 11 / 3 - 2 / 5 = 55 / 15 - 6 / 15 = 49 / 15 = 3.26 kg is the mass of biscuits left.
The mass of biscuits left after Jason eats 2/5 of the packet is approximately 0.8 kg.
To find the mass of the biscuits left after Jason eats 2/5 of the packet, we need to calculate the mass of the biscuits he ate and then subtract it from the initial mass of the packet.
Given:
Mass of the packet of biscuits = 1 1/3 kg
First, let's convert 1 1/3 kg to an improper fraction:
1 1/3 = 3/3 + 1/3 = 4/3 kg
Now, to find the mass of the biscuits Jason ate, we can use the fact that he ate 2/5 of the packet:
Mass of biscuits Jason ate = (2/5) * (4/3) kg
Now, perform the multiplication:
Mass of biscuits Jason ate = (8/15) kg
To find the mass of biscuits left, we subtract the mass Jason ate from the initial mass of the packet:
Mass of biscuits left = 4/3 kg - 8/15 kg
To subtract fractions, we need to find a common denominator, which is 15 in this case:
Mass of biscuits left = (20/15) kg - (8/15) kg
Now, perform the subtraction:
Mass of biscuits left = 12/15 kg
Finally, let's simplify the fraction:
Mass of biscuits left = (12/15) * (1/3) kg
Mass of biscuits left ≈ 0.8 kg
Therefore, the mass of biscuits left after Jason eats 2/5 of the packet is approximately 0.8 kg.
For more details of mass :
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cindy had a party. She invited two guests. Her guests each invited four guests, and then those guests each invited three guests.
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How can you distinguish the base of an isosceles triangle from a leg
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The sum of two numbers is 95. If the larger number is increased by twice the smaller number, the result is 120. What is the larger number?
If S = the smaller number and L = the larger number, then which of the following systems of equations represents the word problem?
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2s+L=120
so it's 120
Answer: correct answer is S + L = 95 and 2S + L = 120
Step-by-step explanation:
There is a sales tax of $6 on an item that cost $80 before tax. The sales tax on a second item is $11.25. How much does the second item cost before tax?
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For an arithmetic sequence where a1 = 13 and the common difference is 3, find s7.
A. 162
B. 157
C. 154
D. 151
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s7 means n=7
a7=7/2[2(13)+(7-1)(3)]=7/2(44)=154
Ans:C
Answer:
C. 154
Step-by-step explanation:
Here we have given that,
First Term a₁ = 13
Common Difference d = 3
Term (n ) = 7
so, we have arithmetic sequence: 13, 16, 19, 22,... and so on
Thus Sum of first 7 terms = [tex]\frac{n}{2} [2a+(n-1)d][/tex]
= 7÷2 [2*13 + (7 - 1)*3 ]
= 3.5 [ 26 + 21 ]
= 154
Based on the lines 5 and 6, what is the best meaning of treasure divine? precious item inconsequential object frivolous amulet incredible wealth
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Johnny is reading a novel for the read-a- Thon. The ratio of the number of pages she read to the pages that remain to be read is 7:3. If the book contains 500 pages in all, how many pages did Johnny reed and how many pages remain left?
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What does it mean for data on a scatter plot to have no association? How does that affect the line of best fit?
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For instance, let's say that I was going to plot (x,y) where x represented somebody's first name, and y represented how old they were
Does a person's name determine their age? Of course not, so there would be no association, or CORRELATION, which means there would not be a way to put in a line of best fit for this.
Hope this helps.
Brian
Answer:
If there isn't any association, or CORRELATION, then you'd have a very difficult time determining a line of best fit since the independent and dependent variables wouldn't have any form of relationship
Step-by-step explanation:
hich is an example of a proportional relationship?
A.The yogurt shop charges .30 per pound.
B.The yogurt shop charges .30 per pound plus $1.00 for the cup.
C.John ran 2 miles in 15 minutes then ran 3 miles in 20 minutes.
D.The car travelled 40 mph for half an hour and then travelled 60 mph for 2 hours
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HELP PLEASE:
What is the point-slope form of the line with slope 25 that passes through the point (−4, −7)?
y−7=2/5(x−4)
y+7=2/5(x+4)
y+4=2/5(x+7)
y−4=2/5(x−7)
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slope(m) = 2/5
(-4,-7)....x1 = -4 and y1 = -7....pay attention to ur signs
now we sub
y - (-7) = 2/5(x - (-4)...not done yet
y + 7 = 2/5(x + 4) <===
Answer:
Correct option is:
y+7=2/5(x+4)
Step-by-step explanation:
The point-slope form of the line with slope m and passing through the point (x1,y1) is given by:
y-y1=m(x-x1)
Here, we are given line with slope m=2/5
and passing through (x1,y1)=(-4,-7)
i.e. x1= -4 and y1= -7
[tex]y-(-7)=\dfrac{2}{5}(x-(-4))[/tex]
[tex]y+7=\dfrac{2}{5}(x+4)[/tex]
Hence, Correct option is:
y+7=2/5(x+4)
Please put down the steps?
Answers
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Explanation:
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Given:
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" √(x + 1) − 1 = x " ; Solve for "x" ;
First, let us assume that "x ≥ -1 "
Add "1" to EACH SIDE of the equation:
→ √(x + 1) − 1 + 1 = x + 1 ;
to get:
→ √(x + 1) = x + 1 .
Now, "square" EACH side of the equation:
→ [√(x + 1) ]² = (x + 1 )² ;
to get:
x + 1 = (x + 1)²
↔ (x + 1)² = (x + 1) .
Expand the "left-hand side" of the equation:
→ (x + 1)² = (x + 1)(x +1) ;
Note: (a+b)(c+d) = ac +ad + bc + bd ;
As such: (x + 1)(x + 1) = (x*x) + (x*1) +(1(x) + (1*1) ;
= x² + 1x + 1x + 1 ;
= x² + 2x + 1 ;
Now, substitute this "expanded" value, and bring down the "right-hand side" of the equation; and rewrite the equation:
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" (x + 1)² = (x + 1)(x +1) " ;
→ Rewrite as: " x² + 2x + 1 = x + 1 " ;
Subtract "x" ; and subtract "1" ; from EACH SIDE of the equation:
→ x² + 2x + 1 - x - 1 = x + 1 - x - 1 ;
to get: → x² − x = 0
Factor out an "x" on the "left-hand side" of the equation:
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x² − x = x(x − 1) ;
→ x (x − 1) = 0 ;
We have: "x" and "(x − 1)" ; when either of these two multiplicands are equal to zero, then the "right-hand side of the equation equals "zero" .
So, one value of "x" is "0" .
The other value for "x" ;
→ x − 1 = 0 ;
Add "1" to each side of the equation:
→ x − 1 + 1 = 0 + 1 ;
→ x = 1 ;
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So, the answers:
" x = 0, 1 " .
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