Cynthia wants to buy a rug for a room that is 18ft wide and 28ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 264 square feet of carpeting. What dimensions should the rug have ?
Answers
SOLUTION
Let us use a diagram to illustrate the information, we have
Now, from the diagram, let the length of the uniform strip of floor around the rug be x, So, this means the length and width of the rug is
[tex]\begin{gathered} \text{length = 2}8-x-x=28-2x \\ \text{width = }18-x-x=18-2x \end{gathered}[/tex]Now, since she can afford to buy a rug of 264 square feet for carpeting, this means that the area of the rug is 264, hence we have that
[tex]\begin{gathered} \text{area of rug = (2}8-2x)\times(18-2x) \\ 264=\text{(2}8-2x)(18-2x) \\ \text{(2}8-2x)(18-2x)=264 \end{gathered}[/tex]Solving for x, we have
[tex]\begin{gathered} \text{(2}8-2x)(18-2x)=264 \\ 504-56x-36x+4x^2=264 \\ 504-92x+4x^2=264 \\ 4x^2-92x+504-264=0 \\ 4x^2-92x+240=0 \end{gathered}[/tex]Dividing through by 4 we have
[tex]\begin{gathered} x^2-23x+60=0 \\ x^2-20x-3x+60=0 \\ x(x-20)-3(x-20)=0 \\ (x-3)(x-20)=0 \\ x=3\text{ or 20} \end{gathered}[/tex]So from our calculation, we go for x = 3, because 20 is large look at this
[tex]\begin{gathered} \text{From the length which is (2}8-2x) \\ 28-2(20) \\ =28-40=-12 \end{gathered}[/tex]length cannot be negative, so we go for x = 3.
Hence the dimensions of the rug becomes
[tex]\begin{gathered} \text{(2}8-2x) \\ =28-2(3) \\ =28-6=22 \\ \text{and } \\ 18-2x \\ 18-2(3) \\ 18-6=12 \end{gathered}[/tex]So the dimension of the rug should be 22 x 12 feet
Related Questions
hello I don't know if you can help me with this but I no am doing something wrong. because at the bottom its not spelling right
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8. The difference of three and a number means x+3 or x-3 because in both equations you have three units plus or minus the number X.
10. '"4 times the sum of a number and three" means
[tex]4\cdot(x+3)=4x+12[/tex]Letter D
Determine the a coordinates of the critical points/numbers for the function f(x)= x/x^2+5
○ x=0, x= -√5, and x = √5
○ x=0
○ No critical points
○ x = √5
○x= -√5 and x = √5
Answers
The critical points for the given function f(x) are -√5 and √5.
so option d is the correct answer.
What is the critical point?A critical point is the part of the domain of a function where the derivative is either equal to zero or the function is not differentiable.
Differentiate the given function f(x)=x/(x²+5)
f'(x)=((x²+5)-x(2x))/(x²+5)²
Using the Quotient Rule for differentiation.
What is Quotient Rule?A method for finding the derivative or differentiation of a function that is given as a ratio or division of two differentiable functions in calculus is known as the quotient rule.
We get f'(x)=5-x²/(x²+5)²
So the derivative is zero at -√5 and √5 and non differentiable at -√5
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hi help I've been trying to solve this for an hour and I just really need the correct answer please help
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First we can se the points that each line passes, and those are:
(-1, 5) & (0, 2)
(-5, -2) & (0, -4)
From this, we calculate each function, that is:
*Line 1:
[tex]m_1=\frac{2-5}{0-(-1)}\Rightarrow m_1=-3[/tex]And we calculate the first function:
[tex]y-2=-3(x-0)\Rightarrow y=-3x+2[/tex]*Line 2:
[tex]m_2=\frac{-4-(-2)}{0-(-5)}\Rightarrow m_2=-\frac{2}{5}[/tex]And we calculate the second function:
[tex]y+4=-\frac{2}{5}(x-0)\Rightarrow y=-\frac{2}{5}x-4[/tex]So the system is:
Solve the inequality and graph the solution set.3 ≤ 4x + 1 < 9
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Okay, here we have this:
Considering the provided inequality, we are going to solve it and graph the solution set, so we obtain the following:
3 ≤ 4x + 1 < 9
3 -1≤ 4x + 1 -1< 9-1
2 ≤ 4x < 8
2/4 ≤ 4x/4 < 8/4
1/2 ≤ x < 2
In interval notation the solution set will be: [1/2, 2)
And if we plot this solution interval we get:
Where the solution set will be the purple part.
Triangle LMN is drawn with vertices at L(−2, 1), M(2, 1), N(−2, 3). Determine the image vertices of L′M′N′ if the preimage is rotated 90° clockwise. L′(1, 2), M′(1, −2), N′(3, 2) L′(−1, 2), M′(−1, −2), N′(−3, 2) L′(−1, −2), M′(−1, 2), N′(−3, −2) L′(2, −1), M′(−2, −1), N′(2, −3)
Answers
ANSWER
L'(1, 2), M'(1, -2), N'(3, 2)
EXPLANATION
The rule for rotating a point (x, y) 90° clockwise is,
[tex](x,y)\rightarrow(y,-x)[/tex]So, the vertices of triangle LMN will be mapped to,
[tex]\begin{gathered} L(-2,1)\rightarrow L^{\prime}(1,2) \\ M(2,1)\rightarrow M^{\prime}(1,-2) \\ N(-2,3)\rightarrow N^{\prime}(3,2) \end{gathered}[/tex]Hence, the image has vertices L'(1, 2), M'(1, -2), N'(3, 2).
A construction crew is lengthening a road. Let y represent the total length of the road (in miles). Let x represent the number of days the crew has worked. Suppose that x and y are related by the equation y=53 + 2x. Answer the questions below. Note that a change can be increased or a decrease. For an increase, use a positive number. For a decrease, use a negative number. What was the road’s length when the crew started working?What is the change per day in the road’s length?
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Given the equation:
[tex]y=53+2x[/tex]Where y represents the total length of the road (in miles), and x represents the number of days the crew has worked.
(a) What was the road’s length when the crew started working?
When the crew started working we have x = 0. Then:
[tex]\begin{gathered} y=53+2\cdot0 \\ \therefore y=53\text{ miles} \end{gathered}[/tex]The road's length is 53 miles.
(b) What is the change per day in the road’s length?
The change per day in the road's length is 2 miles/day.
which is the BEST first step in order to solve this equation15 + 2/3 a = -5a.subtract 15 from both sides b.subtract 2/3 feom both sides c.add 5 to both sides d.multiply by 3 on both sides
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In order to solve this equation, we need to isolate the variable a in one side of the equation.
Since we have the number 15 in the same side of the variable, the best first step would be removing this number 15 from this side, and we do this by subtracting 15 from both sides.
Therefore the answer is a.
What is the slope: (-2, 1) (5,-2)
Answers
Answer: slope = -3/7
Step-by-step explanation:
m(slope) = (y2-y1)/(x2-x1)
m = (-2+-1)/(5--2)
m = (-2-1)/5+2)
m = -3/7
A car used 15 gallons of gasoline when driven 315 miles. Based on this information, which expression should be used to determine the unit rate of miles per gallon of gasoline?
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Given trhat a car used 15 gallons of gasoline to cover 315 miles.
The expression that will be used to determine the unit rate of miles per gallon of gasoline is:
[tex]\frac{315\text{ miles}}{15\text{ gallons}}[/tex]ANSWER:
[tex]\frac{315\text{ miles}}{15\text{ gallons}}[/tex]Exercise 2 Find a formula for Y in terms of X
Answers
Given:
y is inversely proportional to square of x.
The equation is written as,
[tex]\begin{gathered} y\propto\frac{1}{x^2} \\ y=\frac{c}{x^2}\ldots\ldots\ldots c\text{ is constant} \end{gathered}[/tex]Also y = 0.25 when x = 5.
[tex]\begin{gathered} y=\frac{c}{x^2} \\ 0.25=\frac{c}{5^2} \\ 25\times0.25=c \\ c=\frac{25}{4} \end{gathered}[/tex]So, the equation of y interms of x is,
[tex]y=\frac{25}{4x^2}[/tex]When x increases,
[tex]\begin{gathered} \lim _{x\to\infty}y=\lim _{x\to\infty}(\frac{25}{4x^2}) \\ =\frac{25}{4}\lim _{x\to\infty}(\frac{1}{x^2}) \\ =0 \end{gathered}[/tex]Hence, the value of x increases then y decreases.
A food safety guideline is that the mercury in fish should be below one part per million (ppm). listed below are the amounts of mercury found in tuna sushi sampled at different stores in a major city. construct a 98% confidence interval estimate of the mean amount of mercury in the population. does it appear that there’s too much mercury in tuna sushi?0.58 0.82 0.10 0.88 1.32 0.50 0.92
Answers
The amounts of mercury found in tuna sushi sampled at different stores are:
0.58, 0.82, 0.10, 0.88, 1.32, 0.50, 0.92
Number of samples, N = 7
[tex]\begin{gathered} \text{The mean, }\mu\text{ = }\frac{0.58+0.82+0.10+0.88+1.32+0.50+0.92}{7} \\ \mu\text{ = }\frac{5.12}{7} \\ \mu\text{ =}0.73 \end{gathered}[/tex]Standard deviation
[tex]\begin{gathered} \sigma\text{ = }\sqrt[]{\frac{\sum ^{}_{}{(x_1-\mu)^2}}{N}} \\ \sigma\text{ = }\sqrt[]{\frac{(0.58-0.73)^2+(0.82-0.73)^2+(0.10-0.73)^2+(0.88-0.73)^2+(1.32-0.73)^2+(0.50-0.73)^2+(0.92-0.73)^2}{7}} \\ \sigma\text{ =}\sqrt[]{\frac{0.9087}{7}} \\ \sigma\text{ =}\sqrt[]{0.1298} \\ \sigma\text{ = }0.36 \end{gathered}[/tex]The confidence interval is given by the equation:
[tex]\begin{gathered} CI\text{ = }\mu\pm z\frac{\sigma}{\sqrt[]{N}} \\ CI=0.73\pm2.33(\frac{0.36}{\sqrt[]{7}}) \\ CI\text{ = }0.73\pm0.32 \\ CI\text{ = (0.73-0.317})\text{ to (0.73+0.317)} \\ CI\text{ = }0.413\text{ < }\mu<1.047 \end{gathered}[/tex]Please tell me if these are correct if theyre not please help and tell me which ones are the right answers
Answers
Answer:
They're correct
Step-by-step explanation:
114. If plane X averages 800 mph and plane Y averages 400 mph, how manyhours will plane X travel before it overtakes plane Y if plane Y has a 2 hourand 30 minute head start?a.1b. 2c. 5d. 72
Answers
To determine the time taken for the plane X to travel:
If plane X averages 800 mph and plane Y averages 400 mph
Distance column is found by multiplying the rate
by time.
The time taken for plane Y to travel = 2hr 30 minutes = 2.5hrs head start
Be sure to distribute the 400(t +2.5)for Plane Y,
and Plane X to distribute 800t
As they cover the same distance
[tex]\begin{gathered} Dis\tan ce\text{ is equal ,} \\ 800t=400(t+2.5) \\ 800t=400t+1000 \\ 800t-400t=1000 \\ 400t=1000 \\ t=\frac{1000}{400} \\ t=2\frac{1}{2}hr \end{gathered}[/tex]Therefore the plane X will travel for 2 1/2 hours
Hence the correct answer is Option B
Can you please help me out with a question
Answers
AS shown in the figure:
The measure of arc RT = 27
The measure of arc FN = 105
The measure of angle FUN will be as follows:
[tex]m\angle\text{FUN}=\frac{1}{2}(105+27)=\frac{1}{2}\cdot132=66[/tex]So, the answer is option C. 66
Find the volume of a cone with a height of 8 m and a base diameter of 12 mUse the value 3.14 for it, and do not do any rounding.Be sure to include the correct unit in your answer.
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The volume V of a cone with radius r and height h is:
[tex]V=\frac{1}{3}\pi r^{2}h[/tex]And the radius is half the diameter. Since this cone has a diameter of 12 m, the radius is:
[tex]r=\frac{12m}{2}=6m[/tex]And the height is 8m. Thus, the volume V is:
[tex]\begin{gathered} V=\frac{1}{3}\pi(6m)^28m \\ \\ V=\frac{\pi}{3}(36m^2)8m \\ \\ V=\frac{\pi}{3}(288)(m^2\cdot m) \\ \\ V=\pi\cdot\frac{288}{3}m^3 \\ \\ V=96\pi m^{3} \end{gathered}[/tex]Now, using 3.14 for π, we obtain:
[tex]\begin{gathered} V=96\cdot3.14m^3 \\ \\ V=301.44m^{3} \end{gathered}[/tex]Therefore, the volume of that cone is 301.44m³.
Cisco Enterprises in Ontario purchased the following in a single month all-inclusive of taxes:
16,000 units of network routers at $79.25 each
Answers
Answer:
1268000
Step-by-step explanation:
16000x79.25=1268000
Which information will prove a quadrilateral is a square?O All 4 sides are congruentO All 4 sides are congruent and all 4 angles are right anglesO All 4 angles are right anglesO Both pairs of opposite sides are parallel
Answers
Answer
All 4 sides are congruent and all 4 angles are right angles
Step-by-step explanation
In a square, all the four sides measure the same (they are congruent). And the four angles are right angles, that is, they measure 90°
f(x) = -2x² + 2x
Find f(-8)
Answers
Answer:
f(-8)= -112
Step-by-step explanation:
-2(-8)^2 + 2(-8)
-2(-64) + 16
-128 + 16
= -112
hope this helped
have a good day :)
Calculating a rate of change
What is the vertical change form Point A to Point B?
What is the horizontal change from Point A to Point B ?
What is the rate of change shown on the graph? Give the answer as a decimal rounded to the nearest tenth, if necessary?
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Hello there. To solve this question, we'll have to remember some properties about rate of change.
Given the points A and B from a line, we want to determine the vertical change and the horizontal change between the points and then, using these values, determine the rate of change of the function (the line passing through them).
For this, we first find the coordinates of the points.
[tex]A=(2,1)\text{ and }B=(4,2)[/tex]The vertical change is the difference between the y-coordinates of the points, hence
[tex]y(B)-y(A)=2-1=1[/tex]The horizontal change is given by the difference between the x-coordinates of the points, therefore
[tex]x(B)-x(A)=4-2=2[/tex]The rate of change of this function is, finally, given by the ratio between the vertical (rise) and horizontal (run) changes of the function:
[tex]\dfrac{1}{2}=0.5[/tex]This is the rate of change of this function.
calculate the slope of a line passing through the given points (5,-2) and (5,-3)
Answers
Given the points (5,-2) and (5,-3), we can find the slope of the line that passes through them with the following formula:
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ \Rightarrow m=\frac{-3-(-2)}{5-5}=\frac{-1}{0} \end{gathered}[/tex]since we have that the slope is not defined, the line that passes through the points (5,-2) and (5,-3) is the vertical line x=5
Real number between 0 and 6 will be picked according to the probability distribution shown in the figure. Regions under the curve are liable with A, B, C, and D. The area of each is shown in the table. Use the figure and table to answer the parts
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Part A
The probability that a real number between 1 and 4 is picked
P=PB+PC
P=0.15+0.50
P=0.65Part B
The probability that a real number between 2 and 6 is picked
P=PC+PD
P=0.50+0.30
P=0.80This is a non graded practice that I am doing. I don’t under these questions 5-11
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7. The intersection of two intersecting lines is a point.
In the given image, we see that lines NQ and ML intersect at point P.
Therefore, the intersection of NQ and ML is P.
A city has a population of 230,000 people. Suppose that each year the population grows by 4.25%. What will the population be after 12 years?
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Answer:
379,001.
Explanation:
The population of the city grows by 4.25%.
This is a constant factor and models an exponential function.
An exponential population function is of the form:
[tex]\begin{gathered} P(n)=P_0(1+r)^t \\ P_o=\text{Initial Population} \\ r\text{ = growth rate} \\ t\text{ =time in years} \end{gathered}[/tex]From the given problem:
[tex]P_0=230,000,r=4.25\%=0.0425,t=12years[/tex]This then gives us:
[tex]\begin{gathered} P(12)=230000(1+0.0425)^{12} \\ =230000(1.0425)^{12} \\ =379,001 \end{gathered}[/tex]The population after 12 years will be approximately 379,001.
Reflects the given the coordinates points across the y - axis
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Answer:
Explanation:
The reflection over the line y = x gives the following transformation of coordinates
[tex](x,y)\to(y,x)[/tex]therefore, for our case the transformation gives
[tex]\begin{gathered} S(-2,5)\to S^{\prime}(5,-2) \\ T(-3,0)\to T^{\prime}(0,-3) \\ U(1,-1)\to U^{\prime}(-1,1)_{} \end{gathered}[/tex]which are our answers!
The graphical representation of a point and its reflection about the line y =x is the following:
The cost of 5 gallons of ice cream has a varianceof 36 with a mean of 36 dollars during the summer.What is the probability that the samplean would differ from the true mean by more than 0.6 dollars if a sample of 107 5-gallon pails is randomly selected? Roundyour answer to four decimal places.
Answers
Given:
[tex]\begin{gathered} Variance=36 \\ mean=36 \end{gathered}[/tex]To Determine: The samplean would differ from the true mean by more than 0.6 dollars
Solution
Please note that standard deviation is the square root of variance
[tex]\begin{gathered} SD=\sqrt{Variance} \\ SD=Standard-deviation \\ SD=\sqrt{36}=6 \end{gathered}[/tex][tex]\begin{gathered} S.E=\frac{SD}{\sqrt{n}} \\ S.E=Standard-Error \\ n=107 \\ S.E=\frac{6}{\sqrt{107}}=0.5800 \end{gathered}[/tex]Please note that Z is the number of SE(standard error away from the mean. Therefore
[tex]\begin{gathered} Z=\frac{0.6}{0.5800} \\ Z=1.0345 \end{gathered}[/tex][tex]P(|Z|<1.0345)[/tex][tex]P(|Z|<1.0345)=1-P(Z<-1.0345)=1-0.1515=0.8485[/tex]Hence the probability is 0.8485
Find the area round to two decimal places as needed
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To find the area of an obtuse triangle you have to multiply the base of the triangle by the vertical height and divide the result by 2 following the formula:
[tex]A=\frac{b\cdot h}{2}[/tex]The base of the triangle is b= 7 miles and the height is h= 8 miles, using these lengths calculate the area as follows:
[tex]\begin{gathered} A=\frac{7\cdot8}{2} \\ A=\frac{56}{2} \\ A=28mi^2 \end{gathered}[/tex]The area of the triangle is 28 square miles.
Rosa needs to build a wall. She has to start the wall with one postand then every 5.75 feet put another post. The wall will be166.75 feet long. How many posts will she need?
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For every 5.75 feet, here is one post.
Determine the number of posts in a wall of 166.75 feet.
[tex]\begin{gathered} p=\frac{166.75}{5.75} \\ =29 \end{gathered}[/tex]So 29 posts needed for the wall.
The cargo of the truck welghs no more than 2,800 pounds.Use w to represent the weight (in pounds) of the cargo.
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We know that
• The truck weighs no more than 2,800 pounds.
This problem is about inequalities.
"no more" indicates an inequality sign, specifically, it shows that we should use "less than or equal to", because this sign indicates the same as the problem do.
Therefore, the expression of the truck weight is
[tex]w\leq2,800[/tex]In a certain chemical, the ratio of zinc to copper is 4 to 11. A jar of the chemical contains 528 grams of copper. How many grams of zinc does it contain.
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If the ratio of zinc to copper is 4 to 11. A jar of the chemical contains 528 grams of copper. Then 192 grams of zinc does it contain
What is Ratio?A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0.
Given that,
In a certain chemical, the ratio of zinc to copper is 4 to 11.
i.e 4:11 or 4 /11
If A jar of the chemical contains 528 grams of copper
We need to find how many grams of zinc does it contain.
Let us consider it as x.
Form a equation,
4/11=x/528
4×528=11x
2112=11x
Divide both sides by 11
192=x
Hence 192 grams of zinc does it contain.
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Express the interval using inequality notation(1,6)
Answers
The interval (1, 6) contains all the real numbers between 1 and 6, not including any of the endpoints.
This can be written in inequality notation as:
x >1 AND x < 6
But there is a shorter way to write the interval by combining both inequalities:
1 < x < 6
