7 times a number is 8 less than the square of that number. Find the negative solution.

Given:

The radius of the circle is 12 units.

The central angle of the shaded region is 90°

We need to determine the arc length of the shaded region.

Arc length:

The arc length of the shaded region can be determined using the formula,

[tex]Arc \ length=(\frac{\theta}{360} ) 2 \pi r[/tex]

substituting [tex]\theta=90[/tex] and r = 12, we get;

[tex]Arc \ length=(\frac{90}{360} ) 2 (3.14)(12)[/tex]

Multiplying the terms, we have;

[tex]Arc \ length=\frac{6782.4}{360}[/tex]

Dividing, we get;

[tex]Arc \ length=18.84[/tex]

Rounding off to the nearest tenth, we get;

[tex]Arc \ length =18.8[/tex]

Thus, the arc length of the shaded region is 18.8 units.

Answer:

D

Step-by-step explanation:

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Answer:

Hector will pay $74.25 interest on the loan.

Step-by-step explanation:

Hector took out a loan of $900 for 18 months at a rate of 5.5% annually.

Principal amount = $900

Rate of interest = 5.5%

Time = 18 months = 1.5 years

Formula for interest :

[tex]I=\frac{P\times R\times T}{100}[/tex]

  [tex]=\frac{900\times 5.5\times 1.5}{100}[/tex]

 = 74.25

Hector will pay $74.25 interest on the loan.

Answer:

There is 8% (P=0.08) that Frances concludes that the new equipment increases the average daily jewelry production when in fact the new equipment has no effect.

Step-by-step explanation:

We have one-sample z-test with a significance level of 0.08 and a power ot the test of 0.85.

In this test, the null hypothesis will state that the new equipment has the same  productivity of the older equipment. The alternative hypothesis is that there is a significative improvement from the use of new equipment.

The probability that Frances concludes that the new equipment increases the average daily jewelry production when in fact the new equipment has no effect is equal to the probability of making a Type I error (rejecting a true null hypothesis).

The probability of making a Type I error is defined by the level of significance, and in this test this value is α=0.08.

Then, there is 8% that Frances concludes that the new equipment increases the average daily jewelry production when in fact the new equipment has no effect.

Answer:

i. The amount made in interest is $65,000.

ii. Yes, because her expenses is lesser that her interest.

Step-by-step explanation:

i. The woman puts $1,000,000 in a savings account with an interest rate of 6[tex]\frac{1}{2}[/tex]%.

Interest rate = 6[tex]\frac{1}{2}[/tex]% = [tex]\frac{13}{200}[/tex]

The amount she would make in interest = [tex]\frac{13}{200}[/tex] × 1,000,000

                                                                   = 65,000

The amount made in interest is $65,000.

ii. The total of her expenses last year is $47,0000 which is lesser than the amount made in inerest.

After her total expenses, should would have a remainder of;

                                   = $65,000 - $47,0000

                                  = $18,000

Therefore, she would be able to live off the interest.

Answer:

value after 5 years = $23,399.91

after 10 years = $27,377.79

Time it takes for the amount to double = 22.07 years

Step-by-step explanation:

For amounts that are compounded continuously, it means that the interest rate is is added to the investment amount at an infinite number of time, and the formula is given as:

A = P [tex]e^{r.t}[/tex], where:

A = Future value

P = present value

e = constant ≈ 2.7183

r = interest rate in decimal form

t = years

Now for value after 5 years;

A = ???

P = $20,000

r = 3.14% = 0.0314

t =  5 years

∴ A = P [tex]e^{r.t}[/tex]

= 20,000 [tex]e^{0.0314*5}[/tex]

= 20,000 × [tex]e^{0.157}[/tex] = 20,000 × 1.169995 = $23,399.91 ( to 2 decimal places)

(Note that the function '[tex]e[/tex]' can be punched directly from the calculator)

value after 10 years;

A = ???

P = $20,000

r = 3.14% = 0.0314

t =  10 years

∴ A = P[tex]e^{r.t}[/tex]

= 20,000 × [tex]e^{0.0314 * 10}[/tex]

= 20,000 × [tex]e^{0.314}[/tex] = $27,377.79 ( to 2 decimal places)

Time it will take to double the original investment;

A = P [tex]e^{r.t}[/tex]

where;

A = 40,000

P = 20,000

r = 0.0314

t =???

40,000 = 20,000 × [tex]e^{0.0314 * t}[/tex]

[tex]\frac{40,000}{20,000} = \frac{20,000}{20,000} * e^{0.0314*t}[/tex] (divide both sides by 20,000)

2 = [tex]e^{0.0314 * t}[/tex]

Next take the natural logarithm of both sides

㏑(2) = ㏑[tex]e^{0.0314 *t}[/tex]      (㏑[tex]e[/tex] = 1; and the exponent can be brought down )

= 0.6931 = 0.0314 × t × 1

∴ t = [tex]\frac{0.06931}{0.0314}[/tex] = 22.07 years ( to 2 decimal places)

Calculate the value of an investment after 5 and 10 years with continuous compounding at a certain interest rate. Determine the time needed for an investment to double using the continuous compounding formula.

When calculating the future value of an investment with compound interest, we can use the formula A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial sum of money), r is the annual interest rate (as a decimal), t is the time the money is invested for in years, and e is the base of the natural logarithm.

Applying this formula, the future value of a $20,000 investment at a 3.14% interest rate compounded continuously for 5 years would be

Answer:

99% confidence interval for the true mean number of words a third grader can read per minute is [35.5 , 35.9].

Step-by-step explanation:

We are given that a sample of 1584 third graders, the mean words per minute read was 35.7. Assume a population standard deviation of 3.3.

Firstly, the pivotal quantity for 99% confidence interval for the population mean is given by;

                           P.Q. = [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex]  ~ N(0,1)

where, [tex]\bar X[/tex] = sample mean words per minute read = 35.7

            [tex]\sigma[/tex] = population standard deviation = 3.3

            n = sample of third graders = 1584

            [tex]\mu[/tex] = population mean number of words

Here for constructing 99% confidence interval we have used One-sample z test statistics as we know about the population standard deviation.

So, 99% confidence interval for the population mean, [tex]\mu[/tex] is ;

P(-2.58 < N(0,1) < 2.58) = 0.99  {As the critical value of z at 0.5% level

                                                  of significance are -2.58 & 2.58}  

P(-2.58 < [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < 2.58) = 0.99

P( [tex]-2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]{\bar X-\mu}[/tex] < [tex]2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ) = 0.99

P( [tex]\bar X-2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\mu[/tex] < [tex]\bar X+2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ) = 0.99

99% confidence interval for [tex]\mu[/tex] = [ [tex]\bar X-2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] , [tex]\bar X+2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ]

                                                 = [ [tex]35.7-2.58 \times {\frac{3.3}{\sqrt{1584} } }[/tex] , [tex]35.7+2.58 \times {\frac{3.3}{\sqrt{1584} } }[/tex] ]

                                                 = [35.5 , 35.9]

Therefore, 99% confidence interval for the true mean number of words a third grader can read per minute is [35.5 , 35.9].

Answer:21 (2 sf)

Step-by-step explanation:

10%0f 520=52

1%0f 520=52 divide by 10 =5.2

5.2x4=20.8

=21 to 2sf

(arrange the data set from least to greatest)

0, 0, 2, 2, 3, 4, 6, 8

(find the median: *the middle number*)

Median: 2.5

Lower quartile: 1

Upper quartile: 5

Interquartile range : upper quartile - lower quartile = answer

Interquartile range: 5 - 1 = 4

So the IQR or interquartile for the following data set is 4.

Answer:

y = 1.8x + 32

or

y = 9/ 5x  + 32

Step-by-step explanation:

Used a slope calculator

Answer:x=(3d-2c)/(a-b)

Step-by-step explanation:

ax+2c=bx+3d

Collect like terms

ax-bx=3d-2c

x(a-b)=3d-2c

Divide both sides by (a-b)

x(a-b)/(a-b) =(3d-2c)/(a-b)

x=(3d-2c)/(a-b)

The linear equation in two variables is the equation of a straight line. The given formula can be rewritten as   x = (bx + 3d - 2c) / a.

A linear equation is a equation that has degree as one.

To find the solution of n unknown quantities n number of equations with n number of variables are required.

The given equation is as follows,

ax + 2c = bx + 3d

It can be simplified and written in terms of x as,

ax + 2c = bx + 3d

=> ax = bx + 3d - 2c

=> x = (bx + 3d - 2c) / a

Hence, by making x as the subject the given formula can be written as,

x = (bx + 3d - 2c) / a.

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Answer:

15

Step-by-step explanation:

Use Pythagorean theorem.

c² = a² + b²

c² = 9² + 12²

c² = 225

c = 15

Answer:

180 ways

Step-by-step explanation:

Aerospace companies (A) = 5

Energy development companies (D) = 3

Electronics companies (E) = 4

Number of aerospace stocks bought (a) = 2

Number of Energy development stocks bought (d) = 2

Number of Electronics stocks bought (e) = 2

The number of ways that the investor can select his six investments from the recommended list is given by the combination of picking two out five aerospace companies, multiplied by picking two out three energy development companies, multiplied by picking two out four electronics companies:

[tex]n=\frac{5!}{(5-2)!2!}*\frac{3!}{(3-2)!2!}*\frac{4!}{(4-2)!2!}\\ n=10*3*6\\n=180\ ways[/tex]

There are 180 ways for the investor to select the six companies.

There are 180 ways for the investor to select the group of six companies from the recommended list of aerospace, energy development, and electronics companies.

To determine the number of ways the investor can select the group of six companies, we need to use combinations. The combination formula is given by C(n, k) = n! / [k!(n-k)!], where n is the total number of items, and k is the number of items to choose.

First, we calculate the ways to choose the aerospace companies:

Next, calculate the ways to choose the energy development companies:

Finally, calculate the ways to choose the electronics companies:

To find the total number of ways to select the group of six companies, multiply the three results:

Answer: she should use 13 pounds of birdseed and 27 pounds of sunflower seeds.

Step-by-step explanation:

Let x represent the number of pounds of birdseed that she should use.

Let y represent the number of pounds of sunflower seeds that she should use.

Angela​ Leinenbachs' pet store wishes to make a 40​-pound mixture of birdseed and sunflower seeds. It means that

x + y = 40

Birdseed costs ​$0.52 a pound and sunflower seeds cost ​$0.82 a pound. If the mixture would sell for ​$0.72 per a pound, then the total cost of the mixture would be 0.72 × 40 = $28.8

The equation would be

0.52x + 0.82y = 28.8- - - - - - - - - -1

Substituting x = 40 - y into equation 1, it becomes

0.52(40 - y) + 0.82y = 28.8

20.8 - 0.52y + 0.82y = 28.8

- 0.52y + 0.82y = 28.8 - 20.8

0.3y = 8

y = 8/0.3 = 27

x + y = 40

x + 27 = 40

x = 40 - 27

x = 13

Answer:

Step-by-step explanation:

The ordered pairs are

The slope is:

Answer is 2

Answer:

4

Step-by-step explanation:

The graph is:

x    y

1/4  2

1/2  4

Slope formula is (y2-y1/x2-x1)

4-2/0.5-0.25

2/0.25

4

Best of Luck!

Answer:

The radius of the silo is 3.49 m.

Step-by-step explanation:

We have,

Height of the cylinder, h = 30 ft

Volume of cylindrical shaped grain silo, [tex]V=1152\ ft^3[/tex]

It is required to find the radius of silo. The formula of volume of cylinder is given by :

[tex]V=\pi r^2h[/tex]

r is radius

[tex]r=\sqrt{\dfrac{V}{\pi h}} \\\\r=\sqrt{\dfrac{1152}{\dfrac{22}{7}\times 30 }} \\\\r=3.49\ m[/tex]

So, the radius of the silo is 3.49 m.

Answer:

Step-by-step explanation:

let the quadratic eq. be ax²+bx+c=0

or x²+b/a x+c/a=0

or x²+b/ax=-c/a

to complete the square add both sides (b/2a)²or b²/4a²

x²+b/a x+(b/2a)²=b²/4a² -c/a

(x+b/2a)²=(b²-4ac)/(4a²)

taking square root

[tex]x+\frac{b}{2a} =\pm \frac{\sqrt{b^2-4ac} }{2a} \\or\\x=-\frac{b}{2a} \pm\frac{\sqrt{b^2-4ac} }{2a} \\or\\x=\frac{-b \pm\sqrt{b^2-4ac} }{2a}[/tex]

now you must know why -b and not +b

Final answer:

The expression ± √(b² - 4ac) in the quadratic formula refers to the discriminant and cannot be rewritten as b ± √(-4ac) because it would incorrectly separate b from the discriminant and alter the expression's meaning, leading to incorrect solutions.

Explanation:

The student is asking about the quadratic formula, which is used to find the solutions to a quadratic equation of the form at² + bt + c = 0. The formula for the solutions is given by:

-b ± √(b² - 4ac) / 2a

The term ± √(b² - 4ac) cannot be rewritten as b ± √(-4ac) because it alters the original expression's meaning. The square root is taken of the entire expression b² - 4ac, which represents the discriminant of the quadratic equation. It provides information on the nature and number of roots of the equation. Separating b from the discriminant before taking the square root would calculate a different value and therefore lead to incorrect solutions.

Example:

When you substitute the given values, say a = 1, b = 0.0211, and c = -0.0211, into the quadratic formula, you will get:

-0.0211 ± √((0.0211)² - 4(1)(-0.0211)) / (2(1))

Changing the formula to b ± √(-4ac) incorrectly suggests that you can separate b from the discriminant and take the square root of -4ac alone, which would not yield a correct solution.

Answer:

5/6

Step-by-step explanation:

2/3 is equivalent to 4/6

4/6 plus 1/6= 5/6

5/6

Hope this helps :)

Answer:

The answer is 5/6

Step-by-step explanation:

Answer:

4

Step-by-step explanation:

[tex]f(x) = - x + 5 \\ \: f(1) = - 1+ 5 = 4 \\ \: f(5) = - 5+ 5 = 0\\ f(1) +f(5) =4 + 0 \\ \huge \red{ \boxed{f(1) +f(5) = 4}}[/tex]

Answer:

Volume of prism is equal to [tex]96[/tex] units

Step-by-step explanation:

let the side of the cube be of one unit.

Then, the area of the base is equal to [tex]16[/tex] connecting cubes

Assuming a square base of the rectangular prism.

Length of one side of the base of rectangular prism [tex]= 4[/tex] units

Similarly, width of the base of rectangular prism [tex]= 4[/tex] units

The height of the rectangular prism is [tex]6[/tex] units

Volume of the prim [tex]=[/tex] Length [tex]*[/tex] Width  [tex]*[/tex] Height

Substituting the given values in above equation, we get -

[tex]V = 4* 4* 6\\V = 96[/tex]

Volume of prism is equal to [tex]96[/tex] units

Final answer:

To find the volume of a rectangular prism, multiply the base area by the height. In this case, the volume of Lily's rectangular prism would be 96 cubic units.

Explanation:

The volume of a rectangular prism is calculated by multiplying the length, width, and height of the prism.

For this specific rectangular prism with a base area of 16 connecting cubes and 6 layers high, we can calculate the volume as follows:

Volume = Base Area x Height

Volume = 16 x 6

Volume = 96 cubic units

Answer:

42 over 100

Step-by-step explanation:

Answer:21/38

Step-by-step explanation:

Answer:

131 inches

Step-by-step explanation:

1 yard = 36 inches

36 x 2 = 72

72 + 59 = 131

c:

The loan with the shortest term, which is 5 days in this case, is likely to have the highest APR because the finance charge would have less time to be spread out.

The loan that is likely to have the highest Annual Percentage Rate (APR) is the one with the shortest term when all other factors are equal. In this scenario, that would be the loan with a term of 5 days. This is due to the fact that APR is calculated by the annualizing the interest and fees associated with a loan, meaning the shorter the term of a loan, the higher the APR. For instance, a $500 loan with a $20 finance charge has a much higher APR over 5 days as compared to 30 days or 90 days as the finance charge would have lesser time to be spread out.

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Answer: (-2, 3)

Step-by-step explanation:

Answer:

The point-slope form of a linear equation is a formula that allows a person to calculate the slope and point of intercept of a line, and then once you calculate the linear equation, you can calculate the x and y coordinate of any point on the line!

1. Choose two points on a line.

2. Indicate the y1 (the y coordinate of the first point) and y2 (the y coordinate of the second point).

3. Indicate the x1 (the x coordinate of the first point) and the x2 (the x coordinate of the second point)

4. Plug the previously identified variables into the slope formula where the slope is equal to (y2-y1)/(x2-x1)

5. Subtract y2 and y1.

6. Subtract x2 and x1.

7. Divide the quantity in #5 by the quantity in #6. This is your slope.

8. Observe the equation: y=mx+b where "m" is the slope calculated in #7.

9. Plug in the slope for "m"

10. Using one of the points identified earlier, plug in y1 into "y" and x1 into "x". Rearrange and solve for b.

11. Then plug in the value "b" into y=mx+b. Make sure to leave the unknown variables "x" and "y", but make sure to still plug in the "m" calculated earlier!

The one-third cubes which are needed to fill the gap in the prism shown below are [tex]976[/tex].

Cube is a [tex]3D[/tex] solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

Here we have,

Length   [tex]=4\frac{2}{3}[/tex]

Breadth  [tex]=4[/tex]

Height    [tex]=2[/tex]

Here, we have the side of cube [tex]=\frac{1}{3}[/tex],

So, Total volume of the prism [tex]=l*b*h[/tex]

                                                 [tex]=2*4*4\frac{2}{3}[/tex]

                                                 [tex]=\frac{112}{3}[/tex]

Volume of a cube [tex]= a^{3}[/tex]

                             [tex]=(\frac{1}{3} )^{3}=\frac{1}{27}[/tex]

Total number of cubes  [tex]= \frac{\frac{112}{3}}{\frac{1}{27} } =1008[/tex]

Volume of given cubes in prism [tex]= 32*\frac{1}{27} =\frac{32}{27}[/tex]

So, the gap left in the prism  [tex]=\frac{112}{3} -\frac{32}{27} =\frac{976}{27}[/tex]

So, the number of cubes required [tex]=\frac{\frac{976}{27} }{\frac{1}{27} }=976[/tex]

Hence, we can say that the one-third cubes which are needed to fill the gap in the prism shown below are [tex]976[/tex].

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Answer: Im 75% sure the answer is A. Its either A or B.

Step-by-step explanation:

Answer:

[tex]0.56749[/tex].

Step-by-step explanation:

We have been given that the wait times in line at a grocery store are roughly distributed normally with an average wait time of 7.6 minutes and a standard deviation of 1 minute 45 seconds. We are asked to find the probability that the wait time is less than 7.9 minutes.

First of all, we will convert 45 seconds into minutes by dividing by 60 as:

[tex]45\text{ Seconds}=\frac{45}{60}\text{ minutes}=0.75\text{ minutes}[/tex]

So 1 minute 45 seconds will be equal to 1.75 minutes.

Now, we will find z-score corresponding to 7.9 minutes using z-score formula.

[tex]z=\frac{x-\mu}{\sigma}[/tex]

z = z-score,

x = Random sample score,

[tex]\mu[/tex] = Mean,

[tex]\sigma[/tex] = Standard deviation.

[tex]z=\frac{7.9-7.6}{1.75}[/tex]

[tex]z=\frac{0.3}{1.75}[/tex]

[tex]z=0.17142\approx 0.17[/tex]

Now we will use normal distribution to find area under normal curve that corresponds to a z-score of 0.17 that is [tex]P(z<0.17)[/tex].

[tex]P(z<0.17)=0.56749[/tex]

Therefore, the probability that wait time is less than 7.9 minutes would be 0.56749.

Answer:

V = [tex]\frac{1}{3}[/tex]π(8 - [tex]\frac{8}{3}[/tex])[tex]\sqrt{\frac{8}{3} }[/tex] = 9.11 [tex]m^{3}[/tex]

r = 4[tex]\frac{\sqrt{3} }{3}[/tex]

h = [tex]\sqrt{\frac{8}{3} }[/tex]

Step-by-step explanation:

Given that the right triangle whose hypotenuse is [tex]\sqrt{8}[/tex]

We know that:

[tex]r^{2} + h^{2} = 8[/tex]

<=> [tex]r^{2} = 8 - h^{2}[/tex]

The volume of the cone is:

V = π[tex]r^{2} \frac{1}{3} h[/tex]

<=> V = π[tex]\frac{1}{3}(8 - h^{2} )h[/tex]

Differentiate w.r.t h

[tex]\frac{dV}{dh}[/tex] = π [tex]\frac{1}{3}[/tex] (8 - [tex]3h^{2}[/tex])

For maximum/minimum: [tex]\frac{dV}{dh}[/tex] = 0

<=> π [tex]\frac{1}{3}[/tex] (8 - [tex]3h^{2}[/tex])  = 0

<=> [tex]h^{2}[/tex] = [tex]\frac{8}{3}[/tex]

<=> h = [tex]\sqrt{\frac{8}{3} }[/tex]

=> [tex]r^{2}[/tex] = [tex]\frac{16}{3}[/tex]

<=> r = 4[tex]\frac{\sqrt{3} }{3}[/tex]

So the volume of the cone is:

V = [tex]\frac{1}{3}[/tex]π(8 - [tex]\frac{8}{3}[/tex])[tex]\sqrt{\frac{8}{3} }[/tex] = 9.11 [tex]m^{3}[/tex]

Answer:

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