Let the cost of a shirt be x, and let the cost of a sweater be y.
The cost of 5 shirts will be 5x and the cost of 4 sweaters will be 4y.
Since it is given that 5 shirts and 4 sweaters cost $267, it follows that:
[tex]5x+4y=267[/tex]This gives the first equation.
The cost of 3 shirts will be 3x and the cost of 5 sweaters will be 5y.
Since it is given that 3 shirts and 5 sweaters cost $259, it also follows that:
[tex]3x+5y=259[/tex]This forms the second equation.
Next, solve equations simultaneously:
[tex]\begin{gathered} 5x+4y=267 \\ 3x+5y=259 \\ \text{Multiply the first equation by 5 and the second equation by 4:} \\ 25x+20y=1335 \\ 12x+20y=1036 \\ \text{Subtract the second equation from the first to get:} \\ 25x-12x+20y-20y=1335-1036 \\ \Rightarrow13x+0=299 \\ \Rightarrow13x=299\Rightarrow\frac{13x}{13}=\frac{299}{13} \\ \Rightarrow x=23 \end{gathered}[/tex]Substitute the x value, x=23 into the first equation to get the value of y.
[tex]\begin{gathered} 5x+4y=267 \\ \Rightarrow5(23)+4y=267\Rightarrow115+4y=267 \\ \Rightarrow4y=267-115\Rightarrow4y=152 \\ \Rightarrow\frac{4y}{4}=\frac{152}{4}\Rightarrow y=38 \end{gathered}[/tex]Hence, the cost of a shirt is $23 and the cost of a sweater is $38.
44. Consider the equation 4 (x-8) + y = 9(x-2).[Part 1]Find an expression for y of the form ax+b expression for y such that the equationhas infinitely many solutions. Is there more than one such solution? Explain yourreasoning using complete sentences.[Part 2]Find an expression for y of the form ax+b expression for y such that the equationhas no real solutions. Is there more than one such solution? Explain yourreasoning using complete sentences.
Answer:
• 1. y=5x+14
,• 2. y=5x+30
Explanation:
Given the equation:
[tex]4(x-8)+y=9\left(x-2\right)[/tex]To answer the given questions, first, make y the subject of the given equation.
[tex]\begin{gathered} 4(x-8)+y=9(x-2) \\ y=9(x-2)-4\left(x-8\right) \\ y=9x-18-4x+32 \\ y=9x-4x-18+32 \\ y=5x+14\cdots(1) \end{gathered}[/tex]Part 1
We want to find an expression of the form y=ax+b such that the equation has infinitely many solutions.
A system of equations has infinitely many solutions if the two equations in the system simplify to the same line.
Thus, we find an equation that is a multiple of the simplified equation (1) above.
Multiply equation 1 all through by 1.
[tex]\begin{gathered} (y=5x+14)\times1 \\ y=5x+14\cdots(2) \end{gathered}[/tex]Thus, the system of equations:
[tex]\begin{gathered} y=5x+14\cdots(1) \\ y=5x+14\cdots(2) \end{gathered}[/tex]This system has infinitely many solutions since they simplify to the same line.
Part 2
We want to find an expression of the form y=ax+b such that the equation has no real solutions.
For a system of equations to have no real solutions, the two lines formed by the equations must be parallel.
This means that they must have the same slope.
In equation 1:
[tex]y=5x+14\cdots(1)[/tex]The slope is 5.
Thus, find another equation that also has a slope of 5.
[tex]y=5x+30\cdots(3)[/tex]The system of equations below has no real solutions.
[tex]\begin{gathered} y=5x+14\operatorname{\cdots}(1) \\ y=5x+30\operatorname{\cdots}(3) \end{gathered}[/tex]
Julie Ran 2 miles today. Her dad ran 2 times that many miles. How many miles (m) did her dad run?
ANSWER
4 miles
EXPLANATION
If Julie's dad ran twice as many miles as Julie ran, and she ran 2 miles, then he ran:
[tex]2\times2\text{miles}=4\text{miles}[/tex]Write10^8/10^0with a single power of 10 using the appropriate exponent rule.
The given expression is:
[tex]\frac{10^8}{10^0}[/tex]Accoring to exponent rule,
[tex]\begin{gathered} \frac{x^m}{x^n}=x^{m-n} \\ \text{Here, m and n are constants} \end{gathered}[/tex]Applying the above exponent rule to the given expression,
[tex]\frac{10^8}{10^0}=10^{8-0}=10^8[/tex]Therefore, 10^8/10^0 expressed as a single power of 10 is,
[tex]10^8[/tex]Monica volunteered to pick up more than 10 pounds of trash along a highway near her neighborhood. On Friday she picked up 2 pounds of trash, and on Saturday she picked up 3 1/2 pounds of trash. If she picks up N pounds of trash on Sunday, which inequality can be used to determine whether Monica reached her goal?A 5 1/2 + N > 10 B 5 1/2 - N < 10C 5 1/2 > 10 + N D 5 1/2 < 10 - N
First of all, we start writing the information of the problem:
Friday 2 pounds
Saturday 3 1/2 pounds
Sunday N pounds
She has to pick up more than 10 pounds of trash, then:
2 + 3 1/2 + N >10
5 1/2 + N > 10
Finally, the answer is the letter A
given the size of a rectangle are 6in and 5in find the perimeter and the area of the rectangle
lenght (L)= 6in
Height (H)= 5in
perimeter = 2L+2 h
6.In physical education class, Ariel shoots free throws and lay-ups. She earns 1 point for eachfree throw she makes and 2 points for each lay-up she makes. The greatest number of pointsthat she can earn is 30. She has to make at least 15 free throws and lay-ups altogether.a. Write a system of inequalities (two inequalities) to describe the constraints.Specify what each variable represents. (4pts)Ib. Name one possible solution to the system of inequalities and explain what itrepresents in that situation. (4pts)
Let's call t the free throw and l the lay-up.
We know that the minimum free throws and lay-ups together must be 15, this means:
[tex]t+l\ge15[/tex]We also know that the maximum points to earn it 30. Because each t earns 1 point and each l earns 2 points, we can right:
[tex]\begin{gathered} 1\cdot t+2\cdot l\le30 \\ t+2l\le30 \end{gathered}[/tex]So, the system of inequalities is:
[tex]\begin{gathered} t+l\ge15_{} \\ t+2l\le30 \end{gathered}[/tex]One possible solution is to make the inequalities two equalities and solve the system:
[tex]\begin{gathered} t+l=15 \\ t+2l=30 \end{gathered}[/tex]To solve, we can substract the first quation from the second:
[tex]\begin{gathered} (t+2l)-(t+l)=(30)-(15) \\ t+2l-t-l=30-15 \\ t-t+2l-l=15 \\ l=15 \end{gathered}[/tex][tex]\begin{gathered} t+l=15 \\ t+15=15 \\ t=15-15 \\ t=0 \end{gathered}[/tex]So, one possible solution is l=15 and t=0, this means that Ariel shoot 15 lay-ups and no free throws, which earns her 30 points.
The height of a projectile launched upward at a speed of 80 feet/second from a heightof 384 feet is given by the function h(t) = - 16t^2+ 80t + 384. How long will it take theprojectile to hit the ground?
It will take the projectile 8 seconds to hit the ground
Here, we want to get the time it will take the projectile to hit the ground
To do this, we need to use the height on the ground. On the ground, the projectile has a height of 0
Thus, we have it that;
[tex]\begin{gathered} 0=-16t^2+80t+384 \\ \text{divide through by }16 \\ 0=-t^2\text{ + 5t + 24} \\ -t^2\text{ + 5t + 24 = 0} \\ -t^2-3t\text{ + 8t + 24 = 0} \\ -t(t+3)+8(t+3)\text{ = 0} \\ (-t+8)(t+3)\text{ = 0} \\ -t\text{ + 8 = 0 or t +3 = 0} \\ t\text{ = 8 or -3} \end{gathered}[/tex]Since time cannot be negative, we have the time as 8 seconds
Trey is riding in a bike race that goes through a valley and a
Answer:
a) -26 ft
b) 3,148 ft lower
Explanation:
We were given the following data set:
Checkpoint 1: 792 feet
Checkpoint 2: -164 feet
Checkpoint 3: 2,446 feet
Checkpoint 4: 3,055 feet
Checkpoint 5: -93
a) The top of a hill rises 138 feet Checkpoint 2
This is represented as:
[tex]\begin{gathered} Altitude=-164+138 \\ Altitude=-26ft \end{gathered}[/tex]Therefore, the altitude of the top of the hill is -26 feet
b) How much lower is Checkpoint 5 than Checkpoint 4
This is shown below:
[tex]\begin{gathered} Distance=3,055-(-93) \\ Distance=3,055+93 \\ Distance=3,148ft \end{gathered}[/tex]Therefore, Checkpoint 5 is 3,148 feet lower than Checkpoint 4
Using f(x), what is the equation that represents g(x)? g(x) = log5(x) – 3 g(x) = log5(x) + 3 g(x) = log5(x – 3) g(x) = log5(x + 3)
Given
To find the value of g(x).
Explanation:
It is clear that,
[tex]f(x)=\log_5(x)[/tex]Then from the given figure,
It is clear that the value of x moves to the right of the x-axis.
That implies,
[tex]g(x)=\log_5(x-3)[/tex]Hence, the answer is option c)
[tex]g(x)=\log_5(x-3)[/tex]it's a picture of the problem if not you won't understand
ANSWER:
Domain and range
(2, -3), (5,0) and (-2,3)
STEP-BY-STEP EXPLANATION:
When a function is inverted, a switch occurs between domain and range.
So the points are reversed, just like this:
[tex]\begin{gathered} (-3,2)\rightarrow(2,-3) \\ (0,5)\rightarrow(5,0) \\ (3,-2)\rightarrow(-2,3) \end{gathered}[/tex]Therefore, the correct answer is:
The domain and range of a function and its inverse switch. The graph pf the inverse of this functions passes through the points (2, -3), (5,0) and (-2,3)
what does mean mean
There are more than one mean. The most used mean is the arithmetic mean (generally referred just as mean), which is the average of a set of numerical values. Other commonly used mean is the geometric mean. The objective of the means is to indicate the central tendency or typical value of a set of numbers.
what is the center and radius of the circle?(x - 6)² + (y + 8)² = 81
The general equation for a circle is:
[tex](x-h)^2+(y-k)^2=r^2[/tex]where (h,k) is the center and r is the radius.
In this case, we have the following:
[tex](x-6)^2+(y+8)^2=81[/tex]Notice that we can write the equation in the following way:
[tex]\begin{gathered} (x-6)^2+(y+8)^2=81 \\ \Rightarrow(x+6)^2+(y-(-8))^2=(9)^2 \end{gathered}[/tex]therefore, the center of the circle is (h,k)=(6,-8) and the radius is r=9
LBN is a 30°-60°-90° triangle and LB = 18 in. Find NB.
In order to calculate the length of NB, let's use the cosine relation of the angle 30°.
The cosine relation is the length of the adjacent side to the angle over the length of the hypotenuse.
So we have:
[tex]\begin{gathered} \cos (30\degree)=\frac{NB}{BL} \\ \frac{\sqrt[]{3}}{2}=\frac{NB}{18} \\ 2\cdot NB=18\sqrt[]{3} \\ NB=9\sqrt[]{3} \end{gathered}[/tex]A=1/2 bh, solve for h a.h=2Abb.h=2a/bc.h=1/2 Abd.h=Ab
Clear h from the equation
[tex]\begin{gathered} A=\frac{1}{2}b\cdot h \\ 2A=b\cdot h \\ \frac{2A}{b}=h \end{gathered}[/tex]A store sells a computer for $800. Customers can choose to receive a 25% discount and pay it offwith a loan at a simple interest rate of 9%, or they can choose to pay the full price and pay it offin 4 years with no interest. If the customer plans to pay it off in 4 years, which option is better?
Determine the price of customer after discount.
[tex]\begin{gathered} 800-\frac{25}{100}\cdot800=800-200 \\ =600 \end{gathered}[/tex]Determine the interest paid by individual over 4 years.
[tex]\begin{gathered} I=\frac{600\cdot4\cdot9}{100} \\ =216 \end{gathered}[/tex]The amount paid by individual at the end of 4 years is,
[tex]\begin{gathered} A=600+216 \\ =816 \end{gathered}[/tex]So after the discount the individual had to pay $816 and with no discount had to pay onle $800 after 4 years.
So individual must choose to pay off the price in 4 years without any discount and no interest.
Use the parabola tool to graph the quadratic function y=-x^2 – 2x + 8Graph the parabola by first plotting its vertex and then plotting a second point on the parabola.
We have that the parabola is the blue one, and the vertex is (1, 7).
Consider this expression.a3 – 7 + 18When a2 and 6-4, the value of the expression isWhat
Answer:
the value of the expression is 5
Step-by-step explanation:
note that the absolute value function always gives a positive value , that is
| - a | = | a | = a
given
[tex]\sqrt{a^3-7}[/tex] + | b | ← substitute a = 2 and b = - 4
= [tex]\sqrt{2^3-7}[/tex] + | - 4 |
= [tex]\sqrt{8-7}[/tex] + | 4 |
= [tex]\sqrt{1}[/tex] + 4
= 1 + 4
= 5
If n = 22 , overline x (x-bar)=41 , and s = 9 , construct a confidence interval at a 80% confidence level. Assume the data came from a normally distributed population Give your answers to one decimal place
Confidence interval is written as
point estimate ± margin of error
In this scenario, point estimate is the sample mean.
From the information given,
sample mean = 41
sample standard deviation, s = 9
sample size = 22
Since the population standard deviation is unknown, we would calculate the margin of error by applying the formula,
margin of error = t x s/√n
where
t is the test score for the 80% confidence. It is gotten from the student's t distribution table. To find t, the first step is to find the degree of freedom, df
df = n - 1 = 22 - 1
df = 21
From the table,
t = 1.323
margin of error = 1.323 x 9/√22
margin of error = 2.53858
Confidence interval is
41 ± 2.5
Lower limit = 41 - 2.5 = 38.5
Upper limit = 41 + 2.5 = 43.5
Thus, the final answer is
38.5 < μ < 43.5
A merchant could sell one model of digital cameras at list price and receive$162 for all of them. If he had three more cameras, he could sell each onefor $9 less and still receive $162. Find the list price of each camera.
GIven data:
A merchant could sell one model of digital cameras at list price and receive $162 for all of them.
If he had three more cameras, he could sell each one for $9 less and still receive $162.
To Find the list price of each camera.
Let Price = P and Quantity = Q
PQ = 162
(P-9)(Q + 3) = 162 ...(1)
From the first equation,
Q=162/P
Then,(P-9)(162/P+3)=162
[tex]\begin{gathered} \mleft(P-9\mright)(\frac{162}{P}+3)=162 \\ 3P-\frac{1458}{P}+135=162 \\ \text{ X P on both sides} \\ P\cdot P-\frac{1458}{P}P+135P=162P \\ 3P^2-1458+135P=162P \\ P=27,\: P=-18 \\ \end{gathered}[/tex]price acnnot be negative so conside P = 27.
[tex]\begin{gathered} \mleft(27-9\mright)\mleft(Q+3\mright)=162 \\ \frac{\left(27-9\right)\left(Q+3\right)}{18}=\frac{162}{18} \\ Q+3=9 \\ Q+3-3=9-3 \\ Q=6 \end{gathered}[/tex]Hence P= 27 and Q=6.
What is the measure of Z in the parallelogram shown?
A.190
B.90
C.150
D.30
In order to solve this exercise it is important to know that a Parallelogram is.
A Parallelogram is a Quadrilateral. It is a closed flat shape that has four sides.
By definition, some of the properties of a Parallelogram are:
1. The lengths of its opposite sides are equal.
2. Its opposite sides are parallel.
3. The opposite angles are congruent (which means that they have equal measure).
In this case you have the Parallelogram WXYZ. You can identify that the angle X and the angle Z are opposite. Therefore:
[tex]m\angle X=m\angle Z[/tex]Knowing the measure of the angle X, you can determine that:
[tex]m\angle Z=30\degree[/tex]The answer is: Option B.
A basketball player has a 50% chance of making each free throw. What is the probability that the player makes at most nine out of eleven free throws?A. 15/16B. 397/2048C. 509/512D. 193/512
Jack has a total of 36 coins, all quarters and nickels. Together they are worth $4.40.How many of each type of coin does he have?
(algebraic solution whit just 1 variable)
Answer:
q=26; n=10.
Step-by-step explanation:
1. if number of the nickels is 'n', then the number of the quaters is '36-n';
2. if together they are 440 cents, then 15*(36-n)+5*n=440;
3. solution of the equation is n=10, then
4. number of quaters is 36-10=26;
5. Finally, number of the quaters = 26, nickels = 10.
You invested $14,000 in two accounts paying 3% and 6% annual interest, respectively. If the total interest earned for the year was 5570, how much was invested at each rate?
Let x be the amount invested at 3%
Then, (14,000-x) will be the amount invested at 6%.
Interest earned at 3% will be:
[tex]\begin{gathered} I=\frac{PRT}{100} \\ P\colon\text{Principal} \\ R\colon\text{Rate} \\ T\colon\text{Time} \\ I=\frac{x\times3\times1}{100} \\ I=\frac{3x}{100}=0.03x \end{gathered}[/tex]Interest earned at 6% will be:
[tex]\begin{gathered} I=\frac{\text{PRT}}{100} \\ I=\frac{(14,000-x)\times6\times1}{100} \\ I=0.06(14,000-x) \end{gathered}[/tex]Total interest earned is given to $570, thus we have:
[tex]\begin{gathered} 0.03x+0.06(14000-x)=570 \\ 0.03x+840-0.06x=570 \\ 0.03x-0.06x=570-840 \\ -0.03x=-270 \\ x=\frac{-270}{-0.03} \\ x=\text{ \$9,000} \end{gathered}[/tex]Amount invested at 6% will be 14,000 - 9,000 = $5,000.
Hence, the amount invested at 3% and 6% are $9,000 and $5,000 respectively
The number of calories per day consumed by adults is normally distributed with a mean of 3350 and with a standard deviation of 122. Which of the following is closest to the percent of adults who eat more than 3500 calories per day?
Let us write out the given data,
[tex]\begin{gathered} \mu=\operatorname{mean}=3350 \\ \sigma=standard\text{ deviation=122} \\ x=3500 \\ z=z-\text{score} \end{gathered}[/tex]Let us now write the formula for Z-score,
[tex]z=\frac{x-\mu}{\sigma}[/tex]Let us solve for z-score,
[tex]\begin{gathered} z=\frac{3500-3350}{122} \\ =\frac{150}{122}=1.2295 \\ z=1.2295 \end{gathered}[/tex]The probablity that the percent of adult who eat more than 3500 will be
[tex]\begin{gathered} Pr(z>1.23)\Rightarrow Pr(0Hence,the percent of adults who eat more than 3500 calories per day is 11%Given the position of the particle, what the position(s) of the particle when it’s at rest
The position function of a particle is given by:
[tex]X\mleft(t\mright)=\frac{2}{3}t^3-\frac{9}{2}t^2-18t[/tex]The velocity function is the derivative of the position:
[tex]\begin{gathered} V(t)=\frac{2}{3}(3t^2)-\frac{9}{2}(2t)-18 \\ \text{Simplifying:} \\ V(t)=2t^2-9t-18 \end{gathered}[/tex]The particle will be at rest when the velocity is 0, thus we solve the equation:
[tex]2t^2-9t-18=0[/tex]The coefficients of this equation are: a = 2, b = -9, c = -18
Solve by using the formula:
[tex]t=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]Substituting:
[tex]\begin{gathered} t=\frac{9\pm\sqrt[]{81-4(2)(-18)}}{2(2)} \\ t=\frac{9\pm\sqrt[]{81+144}}{4} \\ t=\frac{9\pm\sqrt[]{225}}{4} \\ t=\frac{9\pm15}{4} \end{gathered}[/tex]We have two possible answers:
[tex]\begin{gathered} t=\frac{9+15}{4}=6 \\ t=\frac{9-15}{4}=-\frac{3}{2} \end{gathered}[/tex]We only accept the positive answer because the time cannot be negative.
Now calculate the position for t = 6:
[tex]undefined[/tex]An amusement park is creating signs to indicate the velocity of a roller coaster car on certain hills of the most popular ride. A roller coaster gains kinetic energy as itgoes down a hill. The velocity of an object in kilometers per hour (kph) can be determined by V where is the kinetic energy of the object in joules (1)and is the mass of the object in kilograms(kg)A roller coaster car has a mass of 450 kg and the car has a kinetic energy of 22,500 J on the first hill What velocity does the car obtain on the first hill?
Take into account that the kinetic energy of an object with speed v is given by the following formula:
K = 1/2 Mv²
where m is the mass and v the speed.
M = 450 kg
v = ?
K = 2,500 J
solve for v the previous formula and replace the values of the parameters, just as follow:
K = 1/2 Mv² multiply by 2 both sides
2K = Mv² divide by m
2K/m = v² apply square root both sides
√(2K/M) = v
then, you have
v = √(2(22,500 J)/450kg)
v = 10 m/s
Hence, the speed of the object is 10 m/s
PLESS I DONT WANT TO GET BEAT TODAY BY MY MOM PLESSSS HELP ME х у х Note: Figure is not drawn to scale. If x = 5 units, y = 15 units, and h = 8 units, find the area of the parallelogram shown above using decomposition.
We can decomposite the parallelogram into two right triangles and one rectangle. To find the total area we have to calculate the area of each figure and add them.
• Area of the right triangles (At):
[tex]A_t=2\cdot\frac{x\times h}{2}[/tex]where the first number 2 represents the two triangles, x represents the base of the triangles and h the height. Solving for At:
[tex]A_t=x\times h=5\times8=40units^2[/tex]• Area of rectangle (Ar):
[tex]A_r=y\times h[/tex]where y represents the base of the rectangle and h represents the height of the rectangle. Solving for Ar:
[tex]A_r=15\times8=120units^2[/tex]Finally, finding the area of the parallelogram (Ap):
[tex]A_p=A_t+A_r=40+120=160units^2[/tex]Answer: 160 units²
Find the circumference of this circleusing 3 for TT.13C ~ [?]C = 27r
The circumference of a circle is given by the following formula:
[tex]C=2\pi r[/tex]Where r is the radius of the circle.
This circle has a radius of 13, then the circumference is:
[tex]\begin{gathered} C=2\cdot3\cdot13 \\ C=78 \end{gathered}[/tex]The circumference of this circle is 78.
The line plot shows the birth weights, in pounds, of a litter of 8 puppies.What is the difference in weight between the heaviest and lightest puppy?
ANSWER:
1/2 lb
STEP-BY-STEP EXPLANATION:
We can look at the 8 puppies, where the heaviest has a mass of 6/8 pounds and the lightest has a mass of 2/8 pounds.
We calculate the difference between these two weights, like this:
[tex]d=\frac{6}{8}-\frac{2}{8}=\frac{6-2}{8}=\frac{4}{8}=\frac{1}{2}\text{ lb}[/tex]What you mean between the weights is 1/2 of a pound
Hello, I am unsure how to get to the answer, can you include a sign chart as well for the second derivative thank you.
We want to find the intervals of concavity of the function;
[tex]y=-x^4+8x^2-4[/tex]We start by taking the second derivatives;
[tex]\begin{gathered} y^{\prime}=-4x^3+16x \\ y^{\prime}^{\prime}=-12x^2+16 \end{gathered}[/tex]When a function is concave up, the second derivative is positive, thus; we seek the intervals where;
[tex]\begin{gathered} -12x^2+16>0 \\ -12x^2>-16 \\ x^2<\frac{16}{12} \\ x^2<\frac{4}{3} \\ x<\pm\sqrt{\frac{4}{3}} \end{gathered}[/tex]Let's find the points of inflection, this is where the second derivative is zero;
[tex]\begin{gathered} -12x^2+16=0 \\ -12x^2=-16 \\ x^2=\frac{16}{12}=\frac{4}{3} \\ x=\pm\frac{4}{3} \end{gathered}[/tex]The y values will be;
[tex]\begin{gathered} -(\frac{4}{3})^4+8(\frac{4}{3})^2-4=\frac{44}{9} \\ -(-\frac{4}{3})^4+8(-\frac{4}{3})^2-4=\frac{44}{9} \end{gathered}[/tex]Thus, the answers are;
[tex]\begin{gathered} Concave\text{ }up:(-\sqrt{\frac{4}{3}},\sqrt{\frac{4}{3}}) \\ Inflection\text{ }points:(-\sqrt{\frac{4}{3}},\frac{44}{9}),(\sqrt{\frac{4}{3}},\frac{44}{9}) \end{gathered}[/tex]Thus the answer is option D;
This is a sign chart for the second derivative;