Write 2.154 × 10∧5 in decimal form.

By secant line formula, the slope of the line represented in the table is equal to - 2.

In this problem we must determine the slope of a function based on the secant line formula. After a quick glance at the table, we find that the slope formula can be used as Δx and Δy are constant at every pair consecutive points. Then, the slope of the line is:

m = (- 3 - 9)/(9 - 3)

m = - 12/6

m = - 2

By secant line formula, the slope of the line represented in the table is equal to - 2.

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The dimensions of the most economical shed are 9 ft in length, 9 ft in width, and 9 ft in height if the storage shed is to be built in the shape of a box with a square base.

It is defined as a three-dimensional space enclosed by an object or thing.

It is given that:

A storage shed is to be built in the shape of a box with a square base.

It is to have a volume of 729 cubic feet.

Let x be the length of the base

Let y be the height of the box.

V = 729 cubic feet

The base is square:

l = w = x

V = x²y

y =  729/x²

Cost of the base = $8x²

Cost of the roof = 3x²

Cost of the sides = 4(5.50)xy

= 22xy

Total cost

C= 8x² + 3x² + 22xy

C = 11x² + 22x(729/x²)

C = 11x² + 16038/x

Find the first derivative:

dC/dx = 22x - 16038/x²

Equate; dC/dx = 0

22x - 16038/x² = 0

22x = 16038/x²

22x³ =16038

x³ = 729

x = 9

y = 729/(9)² = 9

Length of base = 9 ft

Width of base  = 9 ft

Height of box = 9 ft

Thus, the dimensions of the most economical shed are 9 ft in length, 9 ft in width, and 9 ft in height.

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The conversion of "z = 2(cos(π/3))" in polar form to rectangular form is equal to 1.

A polar coordinate can be defined as a two-dimensional coordinate system, wherein each point on a plane is typically determined by a distance (r) from the pole (origin) and an angle (θ) from a reference direction (polar axis).

In geometry, the relationship between a polar coordinate (r, θ) and a rectangular coordinate (x, y) based on the conversion rules is given by the following polar functions:

a = rcos(θ)    ....equation 1.

b = rsin(θ)     ....equation 2.

Where:

Note: The exact value of cos(π/3) is equal to ½.

Substituting the given parameters into the formula, we have;

z = 2(½)

z = 2/2

z = 1.

In conclusion, we can logically deduce that the conversion of "z = 2(cos(π/3))" in polar form to rectangular form is equal to 1.

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Complete Question:

Convert z = 2(cos(π/3)) in rectangular form

The speed of the train is 150 km/hour.

The distance covered by the train in 3 4/5 hours is 570 km.

The speed of a body is calculated using the formula, speed = distance/time.

The distance covered is calculated using the formula, distance = speed*time.

The time taken by a body is calculated using the formula, time = distance/speed.

In the question, we are asked for the speed of a train, when it covers a distance of 225 km in 1 1/2 hours.

Distance = 225 km.

Time = 1 1/2 hours = 1.5 hours.

Speed = Distance/Time = 225/1.5 km/hour = 150 km/hour.

Now, we are asked to calculate the distance covered by the train in 3 4/5 hours.

Speed = 150 km/hour.

Time = 3 4/5 hours = 3.8 hours.

Distance = Speed*Time = 150*3.8 km = 570 km.

Thus, the speed of the train is 150 km/hour.

The distance covered by the train in 3 4/5 hours is 570 km.

The provided question is incorrect. The correct question is:

"A train at a uniform speed covers a distance of 225 km in 1 1/2 hours. Find the speed of the train and the distance covered in 3 4/5 hours.

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The annual interest rate is 21%

The given parameters are

Loan Amount, P = $300

Interest, I = $63

Number of years, T = 1

The annual interest rate is calculated as

I = PRT

Substitute the known values in the above equation

63 = 300 * R * 1

Evaluate the product

300R = 63

Divide through by 300

R = 21%

Hence, the annual interest rate is 21%

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Answer:

2.25m²

Step-by-step explanation:

if 1in = 2m then 0.5in = 1m

so 0.75in = 1.5m

area = 1.5 x 1.5

a = 2.25m²

Answer:

[tex]x^{2}+\frac{1}{2} x + \frac{33}{16}= 2 + (x + \frac{1}{4})^2[/tex]

Step-by-step explanation:

[tex]x^{2}+\frac{1}{2} x=x^{2}+2 \times \frac{1}{4} \times x +(\frac{1}{4})^2 - (\frac{1}{4})^2[/tex]

[tex]=[x^{2}+2 \times \frac{1}{4} \times x +(\frac{1}{4})^2] - (\frac{1}{16})[/tex]

[tex]=[x + \frac{1}{4}]^2 - (\frac{1}{16})[/tex]

____________________

then

[tex]x^{2}+\frac{1}{2} x = [x + \frac{1}{4}]^2 - (\frac{1}{16}) -2 + 2[/tex]

then

[tex]x^{2}+\frac{1}{2} x = [(x + \frac{1}{4})^2 - (\frac{1}{16}) -2] + 2[/tex]

then

[tex]x^{2}+\frac{1}{2} x = [(x + \frac{1}{4})^2 - (\frac{33}{16}) ] + 2[/tex]

then

[tex]x^{2}+\frac{1}{2} x = (x + \frac{1}{4})^2 - \frac{33}{16} + 2[/tex]

then

[tex]x^{2}+\frac{1}{2} x + \frac{33}{16}= (x + \frac{1}{4})^2 + 2[/tex]

then

[tex]x^{2}+\frac{1}{2} x + \frac{33}{16}= 2 + (x + \frac{1}{4})^2[/tex]

The function that best fits the graph points is; B: y = 65.0778 * 772.9605ˣ

From the given graph, we can see that is parabolic form and as such we can say it is an exponential function.

Looking at the options, only option B is in exponential form and as such, we will take one point on the graph to check if this is the right function.

Let us use the coordinate (0.9, 26000)

y = 65.0778 * 772.9605ˣ

y = 65.0778 * 772.9605^(0.9)

y = 25837.76

This is very close and as such is the correct option.

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Answer:30%

Step-by-step explanation:

10% of 50 is 50/10 so 10% is 5
15/5=3 so its three lots of 10%

3*10%=30%

Answer: -3 ≤ x ≤ -1

Step-by-step explanation:

1 ≤ 3 - 4x ≤ 9

1 + 3 ≤ - 4x ≤ 9 + 3; Add 3 on all sides

4 ≤ -4x ≤ 12

1 ≤ -x ≤ 3; Divide 4 on all sides

-1 ≥ x ≥ -3; Multiply -1 on all sides(FYI: When multiplying or dividing negative numbers in inequalities, make sure to reverse the signs as well)

Answer:

2

Step-by-step explanation:

1≤3-4x≤9

subtract 3

1-3≤3-4x-3≤9-3

-2≤-4x≤6

divide by 2

-1≤-2x≤3

multiply by -1

1≥2x≥-3

or

-3≤2x≤1

divide by 2

[tex]-\frac{3}{2} \leq \frac{2x}{2} \leq \frac{1}{2} \\-\frac{3}{2} \leq x\leq \frac{1}{2} \\[/tex]

length of interval

[tex]=\frac{1}{2} -(\frac{-3}{2} )\\=\frac{1}{2} +\frac{3}{2} \\=\frac{1+3}{2} \\=\frac{4}{2} \\=2[/tex]

[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]

The given number is :

[tex]\qquad❖ \: \sf \: \cfrac{ - 20}{ \sqrt{10} } [/tex]

since 10 isn't a perfect square, root 10 is an irrational number, and therefore the whole number is irrational as well, and we already know that irrational numbers are also part of real numbers.

[tex] \qquad \large \sf {Conclusion} : [/tex]

So the Correct choices are :

Answer: 25 in³

Step-by-step explanation:

We can calculate the volume of the pyramid by first calculating the volume of a prism with the same dimensions, and then dividing by 3 (all pyramids a volume that's [tex]\frac{1}{3}[/tex] of the volume of a prism with the same dimensions).

The volume of a prism is its base area times its height. The base would be a square, so its area is 5², which is 25. The height is 3 inches, making the prism's volume 75 in³.

The volume of the pyramid would be one-third of this value, which is 75[tex]75\div3[/tex] which is 25 in³.

If the mean, median, and mode of the five numbers 5, 7, 8, A, and B are equal, the possible values of A+B are 20, where A is 8, and B is 12.

The measures of central tendency include the mode, the median, and the mean.

Statistically, a central tendency is a typical value for a probability distribution.

The mode's value appears most often in a set of data values.  The median is the middle value of a dataset.  The mean represents the average.

To determine the possible values of A and B, we can start by finding the median value, which is the middle value in ascending order.

Median (Middle value) = 8

Mode = 8

Total value = 40 (8 x 5)

A = 8

B = 12.

Mean = 8 (40/5)

Thus, if the mean, median, and mode of the five numbers 5, 7, 8, A, and B are equal, the possible values of A+B are 20, where A is 8, and B is 12.

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If 2+3/4 hours is equal to 1+5/5 equal length shift then the single shift is 1.5 hours long.

Given that Paolo helped in the community garden for 2 +3/4 hours this week which is 1+5/6 times length shift.

We are required to find the length of a single shift.

First we have to find the proper fraction of the given mixed fractions.

Fraction is combination of numerator and denominator.The number that is present above the division sign is known as numerator and the number that is present below the division sign is known as denominator.

2+3/4=(8+3)/4=11/4

1+5/6=(6+5)/6=11/6

let the length of a shift is x hours.So,

11/4=11*x/6

11/4=11/6 x

66/44=x

x=3/2

x=1.5 hours.

Hence if 2+3/4 hours is equal to 1+5/5 equal length shift then the single shift is 1.5 hours long.

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Answer:

The volume of the solid is  [tex]1,021m^3[/tex]

Step-by-step explanation:

1. Cylinder

For find the volume of a cylinder we use the next formula:

[tex]V_{cylinder} = \pi h r^2 = \pi \cdot 9m\cdot (5m)^2 \approx 706.86m^3[/tex]

Then the volume of the cylinder is equal to [tex]706.86m^3[/tex]

2. Cones

For find the volume of a cone we use the next formula:

[tex]V_{cone} = \pi \frac{h}{3} r^2 = \pi \cdot \frac{6m}{3} \cdot (5m)^2 \approx 157.08m^3[/tex]

However there are two cones so we have to multiply the volume of one cone for two and we get the total volume for the cones which is [tex]314.16m^3[/tex]

3. Sum of the volumes

Finally we sum the volumes of the cylinder and the cones for get the final result

[tex]V_{cylinder} + V_{cones} = 706.86m^3 + 314.16m^3 \approx 1021m^3[/tex]

So approximating the result is [tex]1021m^3[/tex]

For this, let's go through each problem carefully and step-by-step.

According to the question, the rate of change of the temperature of any object that is defined by T, is directly proportional to the difference of T and the temperature of the environment around it, which we'll denote as X.

[tex]\frac{dT}{dt}[/tex][tex]= k (T-X)[/tex]

K is a constant of proportionality here. And the temperature of the surrounding environment is said to be (-18°C). Thus,

[tex]\frac{dT}{dt} = k(T+18)[/tex].

For part A, in order to find the differential equation for T, we need to solve for k. So we separate the variables and then integrate to solve the equation.

[tex]\int\limits{\frac{dT}{T+18} } = \int\limits {k} \, dt[/tex]

[tex]ln(T+18) = kt+c[/tex]

Now thw inital temperature of a pot of chili is 23°C, so at [tex]t = 0, T_0 = 23*C[/tex].

Substituting 23 for T and 0 for t, we have the following:

[tex]ln(23+18) = k(0)+c[/tex]

[tex]ln(41) = c[/tex]

We know the temperature of chili after 2 hours is 7°C, so we know that when [tex]t = 2, T_1 = 7[/tex]

Substituting t for 2, and T for 7, we get:

[tex]ln(7+18) = 2k+ln(41)[/tex]

[tex]ln(25) = 2k + ln(41)[/tex]

Solving for 2k

[tex]2k = ln(25) -ln(41)[/tex]

[tex]2k = ln(\frac{25}{41})[/tex]

[tex]k = \frac{1}{2}ln(\frac{25}{41})[/tex].

Substituting the value of [tex]\frac{dT}{dt} = k (T+18)[/tex], the differential equation obtained is [tex]\frac{dT}{dt} = \frac{1}{2}ln(\frac{25}{41})(T+18)[/tex].

For part B, to find the temperature of the chili after four hours, we first need to solve the above differential equation.

The solution of the differential equation is given by the equation [tex]ln(T+18) = kt+c[/tex]. Substituting the values of k and c, we have:

[tex]ln(T+18) = \frac{1}{2}ln(\frac{25}{41})t+ln(41)[/tex].

Using the above relation, at any time (t), the temperature (T) can be found out in the following.

At [tex]t = 4, T_2 = \phi[/tex]

[tex]ln(T_2+18)=\frac{1}{2}ln(\frac{25}{41})*4+ln(41)[/tex]

[tex]ln(T_2+18)=2ln(\frac{25}{41})+ln(41)[/tex]

[tex]ln(T_2+18)=-0.989 + 3.714[/tex]

[tex]ln(T_2+18)[/tex] ≅ [tex]2.725[/tex]

Solving the natural logarithm,

[tex]T_2+18 = e^{2.725} = 15.256[/tex]

[tex]T_2 =15.256 - 18[/tex]

[tex]T_2 = -2.744[/tex].

So the temperature of the chili after four hours would be -2.744°C approximately.

To find part C in what time the chili would be 10°C, we need to substitute again.

[tex]t = \phi[/tex][tex], T = -10[/tex]

[tex]ln(-10 + 18) = \frac{1}{2}ln(\frac{25}{41})t + ln(41)[/tex]

[tex]ln(8) = \frac{1}{2}ln(\frac{25}{41})t + ln(41)[/tex]

Solving for [tex]\frac{1}{2}ln(\frac{25}{41})t[/tex],

[tex]\frac{1}{2}ln(\frac{25}{41})t = ln(8) - ln(41)[/tex]

[tex]\frac{1}{2}ln(\frac{25}{41})t = ln(\frac{8}{41})[/tex]

[tex]\frac{1}{2}ln(\frac{25}{41})t = -1.634[/tex]

[tex]ln(\frac{25}{41})t[/tex][tex]= -1.634 * 2[/tex]

[tex](-0.494)t=-3.268[/tex]

[tex]t = \frac{-3.268}{-0.494}[/tex]

[tex]t=6.615[/tex] hours, approximately.

Thus, the chili would reach -10°C at around 6.615 hours.

Hope this helped. This took me a long time.

The length of the vine is 21 ft

what is the circumference of a circle?

the circumference is the perimeter of a circle

the perimeter of the circle = 2πr = 3 ft

the vine is 7 times the perimeter

length of vine = 7×3= 21 ft

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If the tree is 20 feet tall with circumference 3 feet then the length of  vince is around 21 feet.

Given the height of tree is 20 feet and the circumference of tree is 3 feet.

We have to find the length of vine.

Circumference is the perimeter of a circular object. Because the trunk of a tree is in shape of circle so the perimeter of the trunk is 2πr.

We have been given the circumference of tree be 3 feet.

Circumference=2πr

Because vine is 7 times the circumference so the length of vine being:

Length of vine=7*3

=21 feet.

Hence if the tree is 20 feet tall with circumference 3 feet then the vince is around 21 feet long.

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Answer:

YES! it's one-to-one function.

Step-by-step explanation:

One to one function or one to one mapping states that each element of one set, say Set (A) is mapped with a unique element of another set, say, Set (B), where A and B are two different sets. It is also written as 1-1. In terms of function, it is stated as if f(x) = f(y) implies x = y, then f is one to one.

Thus, when we look at this given function

X Y

[tex] - 2 \: \: \: \: \: \: 1 \\ - 5 \: \: \: \: \:- 1 \\ 3 \: \: \: \: \: \: - 5 \\ 1 \: \: \: \: \: \: - 2 \\ 0 \: \: \: \: \: \: 5 \\ - 1 \: \: \: \: \: \: 6 \\ - 6 \: \: \: \: \: \: \: 7 \\ \: \: 7 \: \: \: \: \: \: - 6[/tex]

So, when you look at each value of x is attaches with different value of y and there is no repetition of elements means that one element of x is directly attached to one element of y.

Hope it helps!

Answer:

Gradient of the line is choice D.9/5

Step-by-step explanation:

Slope between two points:slope=(y₂-y₁)/(x₂-x₁)

slope(m)=

[tex] \frac{5 - ( - 4)}{3 - ( - 2)} \\ refine \: (m )= \frac{9}{5} [/tex]

The inference is that the author suggest that looking at birds is silly A. to create a sense of doubt in the reader.

It should be noted that that an inference simply means the conclusion that can be deduced based on the information given in a story.

In this case, A Big Year" is a year in which a person attempts to see as many different species of birds as possible within a particular region. For most in North America who participate in a "Big Year," this region is the lower 48 American states, plus Alaska, Canada, and a couple of French islands off the Canadian coast.

The inference is that the author suggest that looking at birds is silly is to create a sense of doubt in the reader. Therefore, the correct option is A.

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Answer:

Step-by-step explanation:

f(x)=ax³

ax³+2=b(x+1)³

ax³+2=b(x³+3x²+3x+1)

(a-b)x³-3bx²-3bx-b=0

Step-by-step explanation:

let the younger person be x and older be y

4x + 9 = 24

4x = 15

x = 3.75

y = 24 - 3.75

y = 20.25

Answer: y = 3x+5, 3, (-1,2)

Step-by-step explanation:

y = mx + c where m is the slope, so we must find the slope

Slope = rise/run = (5-2)/(0-(-1)) = 3

So y = 3x + c

To find c, we have to substitute one of the coordinates on the graph into the equation, (-1,2), the point that is marked

2 = 3(-1) + c

c = 5

So the equation of the graph is y = 3x + 5

If you do in fact mean [tex]f(1)=f(6)=0[/tex] (as opposed to one of these being the derivative of [tex]f[/tex] at some point), then integrating twice gives

[tex]f''(x) = -\dfrac1{x^2}[/tex]

[tex]f'(x) = \displaystyle -\int \frac{dx}{x^2} = \frac1x + C_1[/tex]

[tex]f(x) = \displaystyle \int \left(\frac1x + C_1\right) \, dx = \ln|x| + C_1x + C_2[/tex]

From the initial conditions, we find

[tex]f(1) = \ln|1| + C_1 + C_2 = 0 \implies C_1 + C_2 = 0[/tex]

[tex]f(6) = \ln|6| + 6C_1 + C_2 = 0 \implies 6C_1 + C_2 = -\ln(6)[/tex]

Eliminating [tex]C_2[/tex], we get

[tex](C_1 + C_2) - (6C_1 + C_2) = 0 - (-\ln(6))[/tex]

[tex]-5C_1 = \ln(6)[/tex]

[tex]C_1 = -\dfrac{\ln(6)}5 = -\ln\left(\sqrt[5]{6}\right) \implies C_2 = \ln\left(\sqrt[5]{6}\right)[/tex]

Then

[tex]\boxed{f(x) = \ln|x| - \ln\left(\sqrt[5]{6}\right)\,x + \ln\left(\sqrt[5]{6}\right)}[/tex]

Answer:

If x^2 = 1, then x = 1 is NOT true because x can also be -1.

Answer:

  305 °F

Step-by-step explanation:

The core temperature of the object after 4 hours can be found using an exponential decay formula to model the decay of the difference between core temperature and ambient.

The solution to the differential equation described by Newton's law of cooling is the exponential equation ...

  y = ab^t +c

where 'a' is the initial core temperature difference from ambient, 'b' is the decay factor of that difference in 1 unit of time period t. 'c' is the ambient temperature.

For this problem, the ambient temperature is c=80, and the differences of interest are ...

  a = 1200 -80 = 1120

  b = (830 -80)/1120 = 75/112

Using these values in the model gives ...

  y = 1120(75/112)^t +80 . . . . . . where y(t) is the core temperature at time t

Note that units of time are hours.

We want y when t=4.

  y = 1120(75/112)^4 +80 ≈ 1120(0.20108) +80 ≈ 305.212

The core temperature after 4 hours is about 305 °F.

__

Additional comment

The differential equation will have a solution of the form ...

  [tex]T-T_R=(T_0-T_R)e^{kt}[/tex]

where k = ln(75/112) ≈ -0.40101

In the above, we defined b = e^k = 75/112. Accuracy with this fraction can be better than using a truncated value of k.

The amount of kilograms of each type of original fruit juice weighs; 12 kg of Apple and 6 kg of grape

The concentrations of apple and grape juice in the two different juice types are given as;

Juice A:

(2/3) apple

(1/3) grape

Juice B:

(1/3) apple

(2/3) grape

Let a be how much juice A is in the mixture;

Let b be how much juice B is in the mix.

Thus;

²/₃a + ¹/₃b = 10     ------(eq 1)

¹/₃a + ²/₃b = 8       ------(eq 2)

Solving both equations simultaneously gives us;

a = 12 and b = 6

Thus, the amount of kilograms of each type of original fruit juice weighs; 12 kg of Apple and 6 kg of grape

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The solution for the simultaneous equations 2x + y - 2z = 12, x + 2y +z =18 and 2x - y + 2z =16 are x = 7, y = 4 and z = 3

Linear equations are equations that have constant average rates of change, slope or gradient

A system of linear equations is a collection of at least two linear equations.

In this case, the system of equations is given as

2x + y - 2z = 12,

x + 2y +z =18

2x - y + 2z =16

Eliminate y and z in the equations by adding the first and the third equation together

This gives

2x + y - 2z = 12,

+

2x - y + 2z =16

--------------------------

4x = 28

Divide both sides by 4

x = 7

Substitute x = 7 in x + 2y +z =18 and 2x - y + 2z =16

7 + 2y +z =18

2(7) - y + 2z =16

This gives

2y + z = 11

-y + 2z= 2

Multiply -y + 2z= 2 by 2

-2y + 4z= 4

Add -2y + 4z= 4 and 2y + z = 11

5z = 15

Divide by 5

z = 3

Substitute z = 3 in -y + 2z= 2

-y + 2*3= 2

Evaluate

y = 4

Hence, the solution for the simultaneous equations 2x + y - 2z = 12, x + 2y +z =18 and 2x - y + 2z =16 are x = 7, y = 4 and z = 3

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