From the problem, we two equations :
[tex]\begin{gathered} y=x^2+4 \\ y=2x+7 \end{gathered}[/tex]Since both equation are defined as y in terms of x, we can equate both equations.
[tex]\begin{gathered} y=y \\ x^2+4=2x+7^{} \end{gathered}[/tex]Simplify and solve for x :
[tex]\begin{gathered} x^2+4=2x+7 \\ x^2-2x+4-7=0 \\ x^2-2x-3=0 \end{gathered}[/tex]Factor completely :
[tex]\begin{gathered} x^2-2x-3=0 \\ (x-3)(x+1)=0 \end{gathered}[/tex]Equate both factors to 0 then solve for x :
x - 3 = 0
x = 3
x + 1 = 0
x = -1
We have two values of x, x = 3 and -1
Substitute x = 3 and -1 to any of the equation, let's say equation 2 :
For x = 3
y = 2x + 7
y = 2(3) + 7
y = 6 + 7
y = 13
One solution is (3, 13)
For x = -1
y = 2x + 7
y = 2(-1) + 7
y = -2 + 7
y = 5
The other solution is (-1, 5)
The answers are (3, 13) and (-1, 5)
Last year, Kevin had $10,000 to invest. he invested some of it in an account that paid 6% simple interest per year, and he invested the rest in an account that paid 10% simple interest per year. after one year, he received a total of $920 in interest. how much did he invest in each account?first account:second account:
Simple interest is represented by the following expression:
[tex]\begin{gathered} I=\text{Prt} \\ \text{where,} \\ I=\text{ interest} \\ P=\text{principal} \\ r=\text{interest rate in decimal form} \\ t=\text{ time (years)} \end{gathered}[/tex]We need to create a system of equations:
Let x be the money invested in the account that paid 6%
Let y be the money invested in the account that paid 10%
So, he received a total of $920 in interest, then:
[tex]920=0.06x+0.1y\text{ (1)}[/tex]And we know that money invested must add together $10,000:
[tex]x+y=10,000\text{ (2)}[/tex]Then, we can isolate y in equation (2):
[tex]y=10,000-x[/tex]Now, let's substitute y=10,000-x in the equation (1):
[tex]\begin{gathered} 920=0.06x+0.1(10,000-x) \\ 920=0.06x+1000-0.1x \\ 0.1x-0.06x=1,000-920 \\ 0.04x=80 \\ x=\frac{80}{0.04} \\ x=2,000 \end{gathered}[/tex]That means, he invested $2,000 in the account that paid 6% simple interest. Now, having x, we are going to substitute x in the second equation (2):
[tex]\begin{gathered} y=10,000-x \\ y=10,000-2,000 \\ y=8,000 \end{gathered}[/tex]He invested $8,000 in the account that paid 10% simple interest per year.
I got stuck and I need help on this I would appreciate the help:0
1) In this problem, we can see that this is an isosceles right triangle.
2) So, one way of solving it is to make use of the Pythagorean theorem. Note that an isosceles triangle has two congruent sides, so we can write out:
[tex]\begin{gathered} a^2=b^2+c^2 \\ b=c \\ 9^2=x^2+x^2 \\ 81=2x^2 \\ 2x^2=81 \\ \frac{2x^2}{2}=\frac{81}{2} \\ x^2=\frac{81}{2} \\ \sqrt[]{x^2}=\sqrt[]{\frac{81}{2}} \\ x=\frac{9}{\sqrt[]{2}} \end{gathered}[/tex]Usually, we rationalize it. But since the question requests the denominator to be a rational one, so this is the answer.
nd the Geometry meand of 4 and 15.
we know that
the geometric mean is the product of all the numbers in a set, with the root of how many numbers there are
so
In this problem we have two numbers
so
the geometric mean is equal to
[tex]\begin{gathered} \sqrt[=]{4\cdot15} \\ \sqrt[]{60} \\ 2\sqrt[]{15} \end{gathered}[/tex]I inserted a picture of the question please state whether the answer is a b c or d PLEASE GIVE A VERY VERY SHORT EXPLANATION
The 30-60-90 triangle is given by
AS we can note , the hypotenuse is twice as long as the shorter leg. Additionally, the longer leg is square root of 3 tines as long as the shorter leg. Therefore, the answer is option C and F
solve for x in the parallelogram below
Which number line shows the solutions to x > 5? O A. A. 3642 8 2 4 6 8 B. 8 -6 -4 -2 0 2 4 6 8 c. -6-4 2 0 2 4 6 8 D. 8 8 4 2 0 2 4 6 8
The answer is option C.
thats where there are intergers greater than 5.
Identify whether the following real world examples should be modeled by a linear quadratic or exponential function
Solution
- Linear:
The general form of a linear function is
[tex]\begin{gathered} y=ax+b \\ where, \\ a,\text{ and b are constants} \end{gathered}[/tex]- Quadratic:
The general form of a quadratic function is:
[tex]\begin{gathered} y=ax^2+bx+c \\ where, \\ a,b,c\text{ are constants} \end{gathered}[/tex]- Exponential:
The general form of an exponential function is:
[tex]\begin{gathered} y=ab^x \\ where, \\ a,b\text{ are constants} \end{gathered}[/tex]- Now that we know the general forms of these functions, we can proceed to solve the question.
- The amount a person is paid per hour in wages is the amount that the person collects for every hour that he works
- Let us imagine that a person receives $a for every hour worked.
- This means that:
After 1 hour, the person makes $a
After 2 hours, the person makes $a + $a = $2a
After 3 hours, the person makes $a + $a +$a = $3a
- We can therefore generalize as follows:
Thus, after x hours, the person makes:
[tex]x\times a=\$ax[/tex]- Thus, the function representing the amount a person makes per hour of work is given by:
[tex]y=ax[/tex]- Comparing this result with the 3 function definitions above, we can see that this corresponds to a Linear function
Final Answer
The answer is Linear
A cash register contains only five dollar and ten dollar bills. It contains twice as many fives as tens and the total amount of money in the cash register is 740 dollars. How many tens are in the cash register?
ANSWER
There are 37 tens in the cash register
EXPLANATION
Given that;
The total amount in the cash register is $740
The cash register contain five dollar and ten dollar
Follow the steps below to find the number of ten dollar in the cash register.
Let x represents the number of $5 and $10 in the cash register.
Recall, that the register contain twice as many $5 as ten dollars and this can be expressed mathematically as
[tex]\text{ 5\lparen2x\rparen+ 10\lparen x\rparen= 740}[/tex]Evaluate x in the above expression
[tex]\begin{gathered} \text{ 10x + 10x = 740} \\ \text{ 20x = 740} \\ \text{ Divide both sides by 20} \\ \text{ }\frac{\text{ 20x}}{\text{ 20 }}\text{ = }\frac{\text{ 740}}{\text{ 20}} \\ \text{ x = 37} \end{gathered}[/tex]Therefore, we have 37 tens in the cash register
The height, in feet, of a particle from the ground is given by the function s(t) = 1.512 + 20r, where 0 ≤ ≤ 17.
Find the velocity of the particle at t = 4.
Answer
feet per second
The velocity is v= 30.6 ft/ sec.
What is a velocity?Velocity defines the direction of the movement of the body or the object. Speed is primarily a scalar quantity. Velocity is essentially a vector quantity. It is the rate of change of distance. It is the rate of change of displacement.
Given that,
We have given the height
s(t) = 0.2[tex]t^{3}[/tex] + 21t, where 0 ≤ [tex]x[/tex] ≤ 17.
To find the velocity we have to differentiate s(t) wrt to t.
s(t) = 0.2[tex]t^{3}[/tex] + 21t
= 0.6[tex]t^{2}[/tex]+21
velocity of the particle at t = 4
s(4) = 0.6*[tex]4^{2}[/tex]+21
= 9.6+21
= 30.6
v= 30.6 ft/ sec
Hence, The velocity is v= 30.6 ft/ sec.
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Find how many years it would take for an investment of $4500 to grow to $7900 at an annual interest rate of 4.7% compounded daily.
To answer this question, we need to use the next formula for compound interest:
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]From the formula, we have:
• A is the accrued amount. In this case, A = $7900.
,• P is the principal amount. In this case, $4500.
,• r is the interest rate. In this case, we have 4.7%. We know that this is equivalent to 4.7/100.
,• n is the number of times per year compounded. In this case, we have that n = 365, since the amount is compounded daily.
Now, we can substitute each of the corresponding values into the formula as follows:
[tex]A=P(1+\frac{r}{n})^{nt}\Rightarrow7900=4500(1+\frac{\frac{4.7}{100}}{365})^{365t}[/tex]And we need to solve for t to find the number of years, as follows:
1. Divide both sides by 4500:
[tex]\frac{7900}{4500}=(1+\frac{0.047}{365})^{365t}[/tex]2. Applying natural logarithms to both sides (we can also apply common logarithms):
[tex]\ln \frac{7900}{4500}=\ln (1+\frac{0.047}{365})^{365t}\Rightarrow\ln \frac{7900}{4500}=365t\ln (1+\frac{0.047}{365})[/tex]3. Then, we have:
[tex]\frac{\ln\frac{7900}{4500}}{\ln(1+\frac{0.047}{365})}=365t\Rightarrow4370.84856503=365t[/tex]4. And now, we have to divide both sides by 365:
[tex]\frac{4370.84856503}{365}=t\Rightarrow t=11.9749275754[/tex]If we round the answer to two decimals, we have that t is equal to 11.97 years.
Simplify cot(t)/csc(t)-sin(t) to a single trig function
The single trig function that simplifies the function is sec(t)
How can we simplify the function?Trigonometry deals with the functions of angles and how they're applied.
Given cot(t)/csc(t)-sin(t)
since csc(t) = 1/sin(t) , we have:
[tex]\frac{ cot(t)}{csc(t)-sin(t)} = \frac{cot(t)}{\frac{1}{sin(t)} - sin(t) }[/tex]
[tex]\frac{ cot(t)}{csc(t)-sin(t)} = \frac{cot(t)}{\frac{1-sin^{2}(t) }{sin(t)} }[/tex]
since:
cos²(t) = 1 - sin²(t)
Therefore we have:
cot(t) / csc(t)-sin(t) = cot(t)/ cos²(t)/sin(t)
cot(t) / csc(t)-sin(t) = cot(t) / cos(t).cos(t)/sin(t)
Since cos(t) / sin(t) = 1/tan(t) = cot(t)
Therefore:
cot(t) / csc(t)-sin(t) = cot(t)/ cot(t)×cos(t)
cot(t) / csc(t)-sin(t) = 1/cos(t)
Since 1/cost = sec(t)
Finally, cot(t) / csc(t)-sin(t) is sec(t).
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If the area of a rectangular field is x2 – 3x + 4 units and the width is 2x – 3, then find the length of the rectangular field.x2- 3 x + 42 x − 3 unitsx2 - 3x + 4 units2x - 3 units3x + 4 units
Solution
We are given the following
[tex]\begin{gathered} Area=x^2-3x+4 \\ \\ Width=2x-3 \\ \\ Length=? \end{gathered}[/tex]Using the Area of a Rectangle we have
[tex]\begin{gathered} Area=lw \\ \\ l=\frac{A}{w} \\ \\ l=\frac{x^2-3x+4}{2x-3} \end{gathered}[/tex]Therefore, the answer is
[tex]\frac{x^{2}-3x+4}{2x-3}units[/tex]what is 9/36 simplified?
Answer:
1/4
Step-by-step explanation:
it can be simplified by dividing both the numerator and denominator with 9.
[tex] \displaystyle \large{ \sf{ \frac{9}{36}}} [/tex]
[tex]\displaystyle \large{ \sf{ \frac{9}{36} = \frac{ \cancel9}{ \cancel3 \cancel6} }}[/tex]
[tex]\displaystyle \large{ \bf{ = \frac{1}{4} }}[/tex]
simplest form is 1/4
I just need to know if You just have to tell me if the circles are open or closed.
Solution
- The solution is given below:
[tex]\begin{gathered} y-2<-5 \\ y-2>5 \\ \\ \text{ Add 2 to both sides} \\ \\ y<-5+2 \\ y<-3 \\ \\ y-2>5 \\ y>5+2 \\ y>7 \end{gathered}[/tex]- Thus, we have:
[tex]\begin{gathered} y<-3 \\ or \\ y>7 \end{gathered}[/tex]- Thus, the plot is:
The distance around a water fountian is 150 inches what is the distance from the edge of the fountian to the center
Answer:
The distance from the edge of the fountain to the centre is approximately 23.87 inches.
The water fountain forms a circle. The distance around the water fountain is the circumference of the circle formed.
Therefore,
circumference = 2πr
150 = 2πr
The distance from the edge of the fountain to the centre is the radius of the circle formed. Therefore,
75 = πr
r = 75 / 3.14159
r = 23.8732616287
r = 23.87 inches
The distance from the edge of the fountain to the centre is approximately 23.87 inches.
Deter mine the intervals for which the function shown below is increasing
Answer:
The interval at which the function is increasing is from x = -2 to x = 0. In interval notation, it is (-2, 0).
Explanation:
See the graph below for the pattern of the function.
As you can see above, from x = -∞ until x = -2, the value of the function decreases from y = +∞ to y = -7.
Then, starting at x = -2 to x = 0, the value of the function increases from y = -7 to y = -3.
Lastly, starting at x = 0 to +∞, the value of the function decreases again from y = -3 to -∞.
Hence, the interval at which the function is increasing is at (-2, 0).
The circle has center O. Its radius is 4 cm, and the central angle a measures 30°. What is the area of the shaded region?Give the exact answer in terms of pi, and be sure to include the correct unit in your answer
Explanation
The area of a portion of a circle with radius 'r' and central angle 'a' in radians is:
[tex]A_{\text{portion}}=\frac{1}{2}\cdot r^2\cdot a[/tex]In this problem, the radius is r = 4cm, and the angle a = 30º.
First we have to express the angle in radians:
[tex]a=30º\cdot\frac{\pi}{180º}=\frac{1}{6}\pi[/tex]And now we can find the area of the shaded region:
[tex]\begin{gathered} A=\frac{1}{2}\cdot(4\operatorname{cm})^2\cdot\frac{1}{6}\pi \\ A=\frac{1}{2}\cdot16\operatorname{cm}^{2}\cdot\frac{1}{6}\pi=\frac{4}{3}\pi \end{gathered}[/tex]Answer
The area of the shaded region is:
[tex]A=\frac{4}{3}\pi cm^{2}[/tex]farm stand has cherries on 2 shelves. Each shelf has 4 boxes. Each box has 8 ounces of cherries. How many ounces of cherries are displayed in all? Write an expression that represents the amount.
64 ounces of cherries are displayed in all in the farm stand.
According to the question,
We have the following information:
Farm stand has cherries on 2 shelves.
Number of boxes in each shelf = 4 boxes
So, the number of boxes in 2 shelves will be (2*4) or 8.
Ounces of cherries in each box = 8 ounces
Now, the ounces of cherries in 8 boxes can be easily found by multiplying the ounces of cherries in 1 box by the number of total boxes.
Ounces of cherries in 8 boxes = (8*8) ounces
Ounces of cherries in 8 boxes = 64 ounces
Now, the expression that represents the amount is (number of shelves*number of boxes*ounces of cherries in each box).
Hence, the ounces of cherries displayed in all is 64 ounces.
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On July 31, Oscar Jacobs checked out of the Sandy Beach hotel after spending four nights. The cost of the room was $76.90 per night. He wrote a check to pay for his four-night stay. What is the total amount of his check, expressed in words?
On July 31, Oscar Jacobs checked out of the Sandy Beach hotel after spending four nights. The cost of the room was $76.90 per night. He wrote a check to pay for his four-night stay. What is the total amount of his check, expressed in words?
step 1
Multiply $76.90 by 4
76.90*4=$307.6
so
expressed in words is
three hundred seven and six tenthsA cake is cut into 12 equal slices. After 3 days Jake has eaten 5 slices. What is his weekly rate of eating the cake?
5
36
35
36
cakes/week
cakes/week
1
1 cakes/week
35
01. cakes/week
4
Answer:
11.2 Slices / Week
Step-by-step explanation:
We know that Jake has eaten 5 slices of cake in 3 days. You can divide 5 / 3 to get an average of 1.6 slices of cake being eaten per day. The question asks what the weekly rate or eating the cake will be, do you need to multiple 1.6 x 7 for the total amount of cake eaten per week, which is 11.2 slices!
Answer:
11.6
explanation
we have 7 days.
7days-3days =4
in 3 days he has eaten 5 slices
again 4-3 days=1
so in 6 days he has eaten 10 slices
we have 1 day left.so if he eats 5 slices in 3 day,how many he eat slices in 1 day?5/3=1.6
10+1.6=11.6
An isosceles right triangle has 6 cm legs . Find the length of the hypotenuse
Step-by-step explanation:
we have a right-angled triangle.
so, we can use Pythagoras
c² = a² + b²
c is the Hypotenuse, a and b are the legs.
in our case
c² = 6² + 6² = 36 + 36 = 72
c = Hypotenuse = sqrt(72) = 8.485281374... cm
Answer:
hypotenuse = √72 (or 8.49)
Step-by-step explanation:
An isosceles right triangle has 6 cm legs . Find the length of the hypotenuse
isosceles right triangle = 2 equal side and 2 equal angleswe use the Pythagorean theorem (In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides)hypotenuse² = 6² + 6²
hypotenuse² = 36 + 36
hypotenuse² = 72
hypotenuse = √72 (or 8.49)
which three statements are true about the line segment CBit's the radius of the circleit is the circumference of the circleit is a cordit is 6cm longit is diameter of the circle it is 7cm longit is 1.75cm long
Answer:
It is the diameter of the circle
it is 7 cm
it is a chord
Explanation:
First, we notice that the line segment CB passes through the centre of the circle and its endpoints touch the circumference - this tells us that CB is the diameter.
Furthermore, any line segment whose endpoints lie on the circumference of the circle is a chord (meaning that the diameter is the longest chord), and so we deduce that CB is also a chord.
Since CB is the diamter, its length is 2 times the radius. The raduis of the circle we know is DA = 3.5 cm; therefore, the dimater is CB = 2 DA = 2 * 3.5 = 7 cm.
Hence, the correct choices are:
It is the diameter of the circle
it is 7 cm
it is a chord
find the slope. A. y= -1/2x - 19/2.
The equation of the line follows the following general structure:
[tex]y=mx+b[/tex]Where m is the slope of the line and b is the y intercept.
Find the corresponding values in the given formula, this way:
In the given equation, m has a value of -1/2, it means the slope is -1/2.
The length of a rectangle is 6 cm more than the width. If the perimeter is 52 cm. What are the dimensions of the rectangle?
LA rectangle has two pairs of sides of the same length. If we call W to the width of the rectangle, we know that the length is 6cm more. If we call L the length of the rectangle:
[tex]L=W+6[/tex]The perimeter of a rectangle is twice the length plus twice the width:
[tex]Perimeter=2L+2W[/tex]Since we know that the perimeter is 52 cm, we can write the system of equations:
[tex]\begin{cases}L={W+6} \\ 2L+2W=52{}\end{cases}[/tex]We can substitute the first equation into the second one:
[tex]2(W+6)+2W=52[/tex]And solve:
[tex]2W+12+2W=52[/tex][tex]\begin{gathered} 4W=52-12 \\ . \\ W=\frac{40}{4}=10\text{ }cm \end{gathered}[/tex]We know that W = 10cm, we can now find L:
[tex]L=10+6=16\text{ }cm[/tex]Thus, the dimensions of the rectangle are:
Length: 16 cm
Width: 10 cm
I have a practice problem in the calculus subject, I’m having trouble solving it properly
The limit of a function is the value that a function approaches as that function's inputs get closer and closer to some number.
The question asks us to estimate from the table:
[tex]\lim _{x\to-2}g(x)[/tex]To find the limit of g(x) as x tends to -2, we need to check the trend of the function as we head towards -2 from both negative and positive infinity.
From negative infinity, the closest value we can get to before -2 is -2.001 according to the values given in the table. The value of g(x) from the table is:
[tex]\lim _{x\to-2^+}g(x)=8.02[/tex]From positive infinity, the closest value we can get to before -2 is -1.999 according to the values given in the table. The value of g(x) from the table is:
[tex]\lim _{x\to-2^-}g(x)=8.03[/tex]From the options, the closest estimate for the limit is 8.03.
The correct option is the SECOND OPTION.
3. Trapezoid JKLM with vertices J(-4, 3), K(-2, 7),L(2,7), and M(3, 3) in the line y = 1.what would the reflection coordinates be
First, we graph the trapezoid and the line
If we reflect the figure across the line y = 1, then we get the following figure
As you can observe in the graph, the vertices would be J'(-4,-1), K'(-2,-5), L'(2,-5), and M'(3,-1).
What value of t makes the following equation true?
5t−2=6t−7
can you please help me. I am running out of time and I really need this grade.
A system of equations is consistent if the system has a solution and it is inconsistent if it has no solution.
Since the lines intersect at a point, the system has a solution and the solution is unique.
If a system has a unique solution, then the system is independent.
Therefore, the given system of equations is consistent and independent. It has a unique solution.
Can you please help me out with the a question
Arc XY = 2π • (PX)/ 4
. = 2π • 5/4
. = 6.28 • 5/4= 31.40/4 = 7.85
Then answer is
Option G) 7.854
find the missing lenghts, the triangle in each pair are similar.
Since the triangles are similar, we have that
[tex]\frac{50}{40}=\frac{x}{52}[/tex]then
[tex]x=\frac{52\times50}{40}=65[/tex]then the answer will be D) 65