A pharmacist has a 6% solution of cough syrup and a 14% solution of the same cough syrup. How many ounces of each must be mixed to make 16 ounces of a 10% solution of cough syrup?

Final answer:

Isaac has 134 square feet of the wall left to paint after subtracting the area he has already painted (28 square feet) from the total area of the wall (162 square feet).

Explanation:

The student's question is regarding an area calculation problem. Isaac is painting a wall with dimensions of 9 feet by 18 feet and has painted a 4 feet by 7 feet section so far. To find the area left to paint, we need to calculate the total area of the wall and subtract the area that's already been painted.

Step 1: Calculate the total area of the wall

The total area of the wall is:

(Length of the wall) × (Width of the wall) = 9 ft × 18 ft = 162 square feet.

Step 2: Calculate the area that has been painted

The area that Isaac has painted is:

(Length of painted section) × (Width of painted section) = 4 ft × 7 ft = 28 square feet.

Step 3: Calculate the area left to paint

To find the remaining area to paint:

(Total area of the wall) - (Area painted) = 162 sq ft - 28 sq ft = 134 square feet.

So, Isaac has 134 square feet of the wall left to paint.

Answer:

Helen: 60mph and Anne: 50mph

Step-by-step explanation:

3.2r+2.8(r-10)=332 is the equation that we use, given the information we have.

We distribute and combine like terms and add 28 to both sides and divide by 6.

3.2r+2.8(r-10)=332

3.2r+2.8r-28=332

                6r=360

             6r/6=360/6

                  r=60  So, Helen's speed is 60mph.

Next, we'll solve Anne's speed.

r-10=50

60-10=50  So, Anne's speed is 50mph.

Anne's average speed was approximately 45 mph, and Helen's average speed was approximately 55 mph.

Let's denote Helen's average speed as "H" and Anne's average speed as "A." We are given that Helen drove for 3.2 hours, and Anne drove for 2.8 hours. We also know that Helen's average speed was 10 miles per hour faster than Anne's, so we can write this relationship as:

H = A + 10

Now, using the formula Speed = Distance / Time, we can express the distances traveled by Helen and Anne:

Distance covered by Helen = H * 3.2

Distance covered by Anne = A * 2.8

Given that the sum of their distances equals the distance between their homes (332 miles):

H * 3.2 + A * 2.8 = 332

Substituting the relationship H = A + 10, we get:

(A + 10) * 3.2 + A * 2.8 = 332

Solving this equation will provide us with Anne's average speed (A), and subsequently, we can find Helen's average speed (H) using the relationship H = A + 10.

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Answer:

The answer is 470 191 764

Step-by-step explanation:

Let's see how we got the figure. First, we need to check our data, or the information supplied.

Data:

       nCr = 1414!/ 3! (1414-3)!

              = 470 191 764

It therefore means that Brian can combine all the 1414 coins in 470 191 764 ways. This makes sense as reflected by the large number of coins he has.  

Answer:

[tex]\sqrt{50}=7.1[/tex]

Step-by-step explanation:

We want to find the square root of 50;

We need to first rewrite 50 as a prime factorization.

[tex]\sqrt{50}=\sqrt{25\times 2}[/tex]

We now split the square root to get:

[tex]\sqrt{50}=\sqrt{25}\times \sqrt{2}[/tex]

Take square root to get:

[tex]\sqrt{50}=5 \sqrt{2}[/tex]

[tex]\sqrt{50}=5(1.414)=7.07[/tex]

To the nearest tenth we have [tex]\sqrt{50}=7.1[/tex]

The constant of proportionality, which is the cost per pair of jeans, is found by dividing the total cost by the number of pairs. For Alyssa's purchase of 3 pairs of jeans for $92.31, the constant of proportionality is $30.77 per pair.

To determine the constant of proportionality for the jeans Alyssa bought, we need to divide the total cost by the number of pairs of jeans she purchased. Since Alyssa bought 3 pairs of jeans for $92.31, we calculate the constant of proportionality as follows:

Therefore, the constant of proportionality is $30.77 per pair of jeans.

Final answer:

The constant of proportionality, or the cost per pair of blue jeans, is $30.77 calculated by dividing the total cost of $92.31 by the number of blue jeans, which is 3.

Explanation:

The student has asked how to find the constant of proportionality, which in this context is the cost per pair of blue jeans when buying multiple pairs. If Alyssa bought 3 pairs of blue jeans for a total of $92.31 on tax free weekend, we can calculate the cost per pair by dividing the total cost by the number of pairs. The constant of proportionality would be: $92.31 / 3 = $30.77 per pair of jeans.

The constant amount of depreciation in the value of boat per year is $ 500

Solution:

When he bought the boat in 2004 he paid $26,500

Therefore,

Initial value in 2004 = $ 26500

In the year 2011, Ryan's boat had a value of $23,000

Value in 2011 = $ 23000

The value of the boat depreciated linearly

If the boat depreciation is linear, then the amount by which the value of boat depreciates must be constant.

Let x be the constant depreciation in the value of boat per year

Then we can say,

Value in 2011 = Initial value in 2004 - nx

Here, "n" is the number of years

2011 - 2004 = 7 years

Therefore,

23000 = 26500 - 7x

7x = 26500 - 23000

7x = 3500

Divide both sides by 7

x = 500

Thus the rate of depreciation per year is $ 500

The annual rate of change of the boat's value is approximately -71.43 dollars.

The annual rate of change of the boat's value can be calculated using the formula for slope of a line. We subtract the initial value from the final value and divide it by the number of years the boat has depreciated. In this case, the initial value is $26,500 and the final value is $23,000. The number of years is 7 (2011 - 2004). So the annual rate of change is ($23,000 - $26,500)/7 = -$500/7 = -71.43. Therefore, the annual rate of change of the boat's value is approximately -71.43 dollars.

The student is asking about the annual rate of change in the value of a boat, which is a problem related to linear depreciation. To solve this, we need to calculate the total amount the boat depreciated over a certain period and then divide by the number of years to get the annual rate.

Ryan's boat was worth $23,000 in 2011 and was purchased for $26,500 in 2004. The total depreciation over these 7 years is $26,500 - $23,000 = $3,500. To find the annual depreciation rate, we divide the total depreciation by the number of years: $3,500 ÷ 7 years = $500 per year.

Therefore, the annual rate of change of the boat's value is $500 per year, which means the boat's value decreased by $500 every year on average.

Answer:

$251,618 is the answer

Step-by-step explanation:

From the previous question, we know he pays $698.94 monthly.

He has to make 360 payments. $698.94 * 360 = $251,618

Answer:

Dimensions are 8 Inches by 14 inches by 20 inches

Step-by-step explanation:

The volume of the box is 2240. This means:

lbh = 2240

Where b is the width.

Width is two inches less than twice the height

b = 2h - 2

length is 4 inches less than 3 times the height

l = 3h - 4

Substituting these parameters into the lbh equation will yield the following:

h * (3h - 4) * (2h - 2) = 2240

h[6h^2-14h+8] = 2240

6h^3 -14h^2 + 8h - 2240= 0

Solving this cubic equation will yield one of the values to be 8. The other two values are complex numbers.

b = 2h - 2

l = 3h - 4

b = 2(8) - 2 = 14

l = 3(8) - 4 = 20

Answer: The value of a is 2 and the value of b is 3.

Step-by-step explanation:

Given : △CDE maps to △STU with the transformations (x, y) → (x − 2, y − 2) →(3x, 3y)

The first transformation is a translation ,so there will be no change in the length of the sides ∵  translation is a rigid motion.

The second transformation is a dilation ,so there will be a change in the length of the sides by scale factor of 3. ∵  dilation is not a rigid motion.

Basically , by combining both transformation:

Length of Side in  △STU = 3 x (Corresponding side in △CDE )

⇒ ST = 3CD   and TU  = 3 DE

If CD = a + 1, DE = 2a − 1, ST = 2b + 3 and TU = b + 6 , then

2b + 3=3(a + 1)  and b + 6 = 3(2a − 1)

⇒ 2b + 3=3a+3 and  b + 6 = 6a-3

⇒ 3a-2b=0        (i)   and b = 6a-9             (ii)

Put value of b from (ii) in (i)  , we get

3a-2(6a-9)=0

⇒ 3a-12a+18=0

⇒ -9a=-18

⇒ a= 2

Put value of a in (ii) , we get

b= 6(2)-9

=12-9=3

Hence, the value of a is 2 and the value of b is 3.

Answer:

a = 4 , b = 6

Step-by-step explanation: I did the same question

Answer:

[tex] \frac{24}{26} = \frac{12}{13} \\ [/tex]

Answer:

A

Step-by-step explanation:

Sin(C) = opposite/hypotenuse

Hypotenuse is the length opposite to the right angle, AC for this triangle

Opposite is the length opposite to the angle, AB in this case

Sin(C) = AB/AC

= 24/26 = 12/13

Answer:

6 baseball games remain in the season.

Step-by-step explanation:

Let the number of baseball games remaining be 'x'.

Given:

Number of innings in 1 baseball game = 9

Number of innings to be played at home = 45

Number of innings to be played away from home = 45

Total number of innings played in the given season = 36

Now, total number of innings to be played in a season is equal to the sum of the innings played at home and away from home.

So, Total number of innings = 45 + 45 = 90 innings.

Now, out of 90 innings, 36 innings are already played.

So, the number of innings that remain is given as:

Innings remaining = Total innings - Innings already over

Innings remaining = 90 - 36 = 54

Now, 9 innings is equivalent to 1 baseball game.

So, 54 innings is equivalent to 'x' baseball games.

Setting up a proportion and solving by cross multiplication, we get:

[tex]\frac{9}{1}=\frac{54}{x}\\\\9x=54\\\\x=\frac{54}{9}=6[/tex]

Therefore, 6 baseball games remain in the season.

Answer:option D is the correct answer

Step-by-step explanation:

The given triangle is a right angle triangle.

From the given right angle triangle,

The hypotenuse of the right angle triangle is 8

With m∠60 as the reference angle,

The adjacent side of the right angle triangle is 4

The opposite side of the right angle triangle is 4√3

To determine Cos 60, we would apply

the cosine trigonometric ratio.

Cos θ = adjacent side/hypotenuse. Therefore,

Cos 60 = 4/8 = 1/2

To determine Tan 60, we would apply the Tangent trigonometric ratio.

Tan θ = opposite side/adjacent side. Therefore,

Tan 60 = 4√3/4

Tan 60 = √3

Answer:

  2 .The slope of Function A is less than the slope of Function B

Step-by-step explanation:

A graph of Function A shows it has a y-intercept of 4, the same as that of Function B. (Statements 3 and 4 are not correct.)

The slope of Function A is 2, which is less than the slope of 3 that Function B has. (Statement 2 is correct; statement 1 is not.)

_____

More detailed working

The slope of Function A can be figured easily between the points with x-values that differ by 1:

  m = (y3 -y2)/(x3 -x2) = (24-22)/(10-9) = 2/1 = 2 . . . . . Fun A has slope of 2.

The slope of Function B is the coefficient of x in the equation: 3.

__

The y-intercept of Function A can be found starting with point-slope form:

  y -22 = 2(x -9)

  y = 2x -18 +22

  y = 2x +4 . . . . . . . slope-intercept form

The intercept of +4 is the same as that of Function B.

Answer:

Step-by-step explanation:

Triangle IGH is a right angle triangle.

From the given right angle triangle

GI represents the hypotenuse of the right angle triangle.

With m∠I as the reference angle,

HI represents the adjacent side of the right angle triangle.

GH represents the opposite side of the right angle triangle.

To determine Hl, we would apply the tangent trigonometric ratio

Tan θ = opposite side/adjacent side. Therefore,

Tan 42 = 11/HI

0.9HI = 11

HI = 11/0.9

HI = 12.22

8! - 6!

Evaluate the factorials.

40320 - 720

Subtract.

39600.

The expression (8!) - (6!) is equal to 39600.

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After the expression has been evaluated, looking at the main value of 8! − 6!, it becomes  39,600. Therefore 8! − 6! = 39,600.

To evaluate the expression 8!−6!, we first need to determine the value of 8! and the value of 6!, and then subtract the two results.

8! (read as "8 factorial") means:

8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320

6! (read as "6 factorial") means:

6 × 5 × 4 × 3 × 2 × 1 = 720

subtract 6! from 8!:

8!−6! = 40,320 − 720 = 39,600

Therefore 8! − 6! = 39,600.

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Answer:

36π cm^2.

Step-by-step explanation:

This is a sphere . Surface area =  4πr^2.

This sphere has surface area = 4π3^2

= 36π.

The surface area of the sphere would be = 36πcm². That is option C.

Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object.

here, we have,

to calculate the surface area of a sphere:

The surface area of a sphere can be calculated through the use of the formula = 4πr²

Where,

radius (r) = 3 cm

surface area

=4πr²

= 4π × 3²

= 36π cm² ( in the terms of π)

Hence, The surface area of the sphere would be = 36πcm². That is option C.

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Answer:

The answer is 16. 8 divided 6 is 1.333. When 1.333 is multipled by 12 you get 15.9.

Answer:

x = 16, y = 4.5, z = 7.5

Step-by-step explanation:

Similar figures have the sides in the same ratio

Ratio = BC/FG = 8/6 = 4/3

AD/EH = 4/3

x/12 = 4/3

x = 16

AB/EF = 4/3

6/y = 4/3

y = 6×3÷4

y = 4.5

DC/HG = 4/3

10/z = 4/3

z = 10×3÷4

z = 7.5

Answer: There are 1540 different orders.

Step-by-step explanation:

                              [tex]\dfrac{n!}{a!\ b!\ ....}[/tex]

Given : Total items = 22

Defective items = 3

Not defective items = 22-3 = 19

Then, the number of different orders can the 22 items be arranged if all the defective items are considered identical and all the non-defective items are identical of a different class :

[tex]\dfrac{22!}{3!\times19!}\\\\=\dfrac{22\times21\times20\times19!}{6\times19!}=1540[/tex]

Hence, there are 1540 different orders.

Answer:

Option C - determine whether [tex]9-4(-x)^2[/tex] is equivalent to [tex]-(9-4x^2)[/tex] or not.

Step-by-step explanation:

To find : Which statement best describes how to determine whether [tex]f(x) = 9-4x^2[/tex] is an odd function?

Solution :

We have a property for odd functions,

Let f(x) be an odd function then it must satisfy

[tex]f(-x)= -f(x)[/tex]

Now, we have been given the function [tex]f(x) = 9-4x^2[/tex]

For this function to be odd, it must satisfy the above property.

Replace x with -x,

[tex]f(-x)=9-4(-x)^2[/tex]

and

[tex]-f(x)=-(9-4x^2)[/tex]

Hence, in order to the given function to be an odd function, we must determine whether [tex]9-4(-x)^2[/tex] is equivalent to [tex]-(9-4x^2)[/tex] or not.

Therefore, C is the correct option.

Answer:

Step-by-step explanation:

Let x represent the maximum number of miles that she can drive.

Riley needs to rent a car while on vacation. The rental company charges $18.95, plus 16 cents for each mile driven. Converting 16 cents to dollars, it becomes 16/100 = $0.16

Assuming Riley drives the car for x miles, the total charge would be

0.16x + 18.95

If Riley only has $50 to spend on the car rental, it means that

0.16x + 18.95 = 50

0.16x = 50 - 18.95

0.16x = 31.05

x = 31.05/0.16 = 194.0625

The maximum number of miles that

she can drive is 194 miles.

By first calculating the Garcia family's travel rate of 409 miles per day, we can estimate it will take them approximately 9 days to travel from San Diego, California to Bar Harbor, Maine.

The subject of this question is Mathematics, and it involves understanding and applying the concept of rate – the speed at which something happens over a particular period of time. In this case, we have the Garcia family traveling from San Diego, California, to Bar Harbor, Maine. They have traveled 2,045 miles in 5 days.

To find out how long it will take them to travel the entire way, we first need to calculate the rate at which they are traveling. We do that by dividing the total distance they have traveled by the total number of days it took them to travel that distance: 2045 miles / 5 days = 409 miles per day.

The distance from San Diego to Bar Harbor is about 3,305 miles. So, if they continue to travel at a rate of 409 miles per day: 3305 miles / 409 miles per day = about 8.08 days. Since they can't travel a fraction of a day, we'll round that up to 9 days. So it will take them approximately 9 days to travel from San Diego to Bar Harbor at their current rate.

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Answer: Polaris would appear at [tex]26\°[/tex] latitude

Step-by-step explanation:

Let's begin by explaining that Latitude is the angular distance between the Earth's equator, and a specific point on the planet. It is measured in degrees and is represented according to the hemisphere in which the point is located, which can be north or south latitude.  

In this sense, latitude [tex]0\°[/tex] refers to the equatorial line that divides the Earth in two hemispheres (North and South), and Miami's latitude [tex]26\°[/tex] refers to the Northern hemisphere.

On the other hand, talking about the North Star (also known as Polaris); if we were just in the North Pole (latitude [tex]90\°[/tex]), Polaris would by exactly over our heads or the zenith ([tex]90\°[/tex] over the horizon), but as we go until latitude [tex]26\°[/tex], Polaris altitude will be approximately at that same angle over the horizon.

Hence, from an observer located in Miami, Polaris would appear at [tex]26\°[/tex] N.

In Miami, with a latitude of 26° N, the North Star appears at an altitude of 26° above the horizon. As you move southward, it appears lower; when you travel north, it appears higher. Precise navigation also considers the slight angular difference between Polaris and the true celestial pole.

In Miami, Florida, which has a latitude of 26° N, the North Star, or north celestial pole, would appear at an altitude of 26° above the northern horizon. This is because the altitude of the North Star above the horizon is roughly equivalent to the latitude of the observer's location in the Northern Hemisphere. Therefore, as one drives southward from Miami to a city at a lower latitude, the North Star would appear lower in the sky. Conversely, driving northward would make the North Star appear higher.

It's important to note that due to the Earth's curvature, as you move southward from Miami, both the North Star and the southern sky would appear to sink, while the opposite would occur as you move northward. If you were to reach the equator, the North Star would align with the northern horizon, and it would not be visible from latitudes south of the equator. For precision in navigation or astronomy, one must also account for the small angular distance between Polaris and the true north celestial pole.

Answer:

  A.  -2

  B.  -10

Step-by-step explanation:

The slope of a perpendicular line will be the negative reciprocal of the slope of the given line:

  -1/(1/2) = -2 . . . . slope of the perpendicular line

__

The y-intercept will let the given point satisfy the equation ...

  y = -2x +b

  2 = -2(-6) +b

  -10 = b . . . . . . . subtract 12. This is the y-intercept.

_____

The graph shows the two lines and the points they go through.

Answer:

$2253.98

Step-by-step explanation:

Jim received a $2000 loan from his bank. The loan accrues 3% interest every 3 months.

[tex]A=P(1+\frac{r}{n} )^{nt}[/tex]

P=2000

r= 3%=0.03  and t= 4 years

interest every 3 months so n= 4

[tex]A=2000(1+\frac{.03}{4} )^{4 \cdot 4}[/tex]

[tex]A=2000(1+\frac{.03}{4} )^{16}\\A=2000(1.0075)^{16}\\\\A=2253.98[/tex]

Compound interest is the addition of interest. The interest that is needed to be paid by Jim in the 4 years of tenure is $2253.98.

Compound interest is the addition of interest on the interest of the principal amount. It is given by the formula,

[tex]A = P(1+ \dfrac{r}{n})^{nt}[/tex]

We know that the Principal amount received by Jim is $2000, while the interest that Jim needs to pay is 3% quarterly, therefore, he needs to pay the interest 4 times a year. Thus, the value of n is 4.

Now, we know all the values therefore, substitute the values in the formula of compound interest,

[tex]A = P(1+ \dfrac{r}{n})^{nt}[/tex]

[tex]A = 2000(1+ \dfrac{3}{4})^{4 \times 4}\\\\A = \$2,253.98[/tex]

Hence, the interest that is needed to be paid by Jim in the 4 years of tenure is $2253.98.

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Answer:

Step-by-step explanation:

The fraction of tickets that are adult tickets is ...

  ((average price per ticket) - (child's ticket cost)) / (difference in ticket costs)

so the fraction of adult tickets is ...

  ((2881/548) -3.50)/(6.50 -3.50) = 321/548

Then the number of adult tickets is ...

  (321/548)·548 = 321

and the number of child tickets is ...

  548 -321 = 227

321 adult and 227 child tickets were sold that night.

_____

If you want to write an equation, you can let "a" represent the number of adult tickets sold. Total revenue is ...

  6.50a +3.50(548 -a) = 2881

  3.00a +1918 = 2881 . . . . . . eliminate parentheses

  3a = 963 . . . . . . . . . . . . . . . subtract 1918

  a = 321 . . . . . . . . . . . . . . . . . divide by 3

The number of child tickets is ...

  548 -a = 548 -321 = 227

Answer:

Step-by-step explanation:

given that a  bacteria culture initially contains cells and grows at a rate proportional to its size.

If P be the size then growth rate

[tex]P'=kP[/tex] where k is constant of proportionality

separate the variables as

[tex]\frac{dP}{P} =kdt\\ln P =kt+C\\P = Ae^{kt}[/tex]

If after 1 hour population is B (say)

[tex]B=Ae^{k} \\\\k = ln B - ln A[/tex]

then k = ln B - ln A

Using this

P(t) = [tex]Ae^{(lnB-lnA)t}[/tex]

b) P(e) = [tex]Ae^{(lnB-lnA)3}[/tex]

c) Rate of growth = [tex](ln B- ln A)Ae^{(lnB-lnA)3}[/tex]

Unless you give B value, d cannot be solved

Answer:

d. −7/32

Step-by-step explanation:

The expression that we have to evaluate in this problem is:

[tex]7x^{-3}y^{-1}[/tex]

We have to evaluate this expression for:

x = -2

y = 4

We start by rewriting the expression by rewriting it using the following:

[tex]a^{-n}=\frac{1}{a^n}[/tex]

So the expression can be rewritten as

[tex]7x^{-3}y^{-1}=\frac{7}{x^3 y}[/tex]

Now we observe that:

[tex](-2)^3=(-2)(-2)(-2)=-8[/tex]

Therefore, by substituting x = -2 and y = 4 into the expression, we find:

[tex]\frac{7}{(-2)^3\cdot 4}=\frac{7}{-8\cdot 4}=-\frac{7}{32}[/tex]

Answer:

Area of good part of roof =

[tex]A(x) = (24x + 144)~ft[/tex]

Step-by-step explanation:

We are given the following in the question:

Length of square roof =

[tex](x+12)~ft[/tex]

Length of square patch =

[tex]x~ft[/tex]

Area of square =

[tex]A = (\text{Side})^2[/tex]

We have to find the area of good part of roof.

Area of good part of roof =

Area of roof - Area of patch

[tex]A(x) = (x+12)^2 - (x)^2 \\A(x) = (x+12+x)(x+12-x)\\A(x) = (2x+12)12\\A(x) = (24x + 144)~ft[/tex]

is the required area of roof.

Answer:

Step-by-step explanation:

Triangle ABC is a right angle triangle.

From the given right angle triangle,

AC represents the hypotenuse of the right angle triangle.

With m∠A as the reference angle,

AB represents the adjacent side of the right angle triangle.

BC represents the opposite side of the right angle triangle.

To determine m∠A, we would apply

the cosine trigonometric ratio.

Cos θ = adjacent side/hypotenuse. Therefore,

Cos A = 13/15 = 0.8667

A = Cos^-1(0.8667)

A = 29.92