Draw Figure 4. How many shaded squares are in Figure 5?
Explain your reasoning

The sample will have a mass of 20mg approximately 67.6 days after it was set aside for observation.

(a) The exponential function M that gives the mass of the sample t days after the original sample was set aside for observation can be expressed as M(t) = 50(0.9875)^t.

The function M(t) is obtained by multiplying the initial mass of the sample (50mg) by the daily decay rate (0.9875, which corresponds to 1 - 1.25%). The exponent t represents the number of days that have passed since the sample was set aside.

(b) To find M(15), we substitute t = 15 into the exponential function M(t):

M(15) = 50(0.9875)^15 ≈ 40.15 mg.

Interpretation: After 15 days, the sample will have a mass of approximately 40.15 milligrams. This means that the sample has experienced a decay of approximately 9.85 milligrams over the 15-day period.

(c) To determine when the sample will have a mass of 20mg, we need to solve the equation M(t) = 20.

Setting M(t) = 20 in the exponential function, we have:

50(0.9875)^t = 20.

Dividing both sides by 50, we get:

(0.9875)^t = 0.4.

To solve for t, we can take the logarithm of both sides. Using the natural logarithm (ln), we have:

ln((0.9875)^t) = ln(0.4).

Applying the logarithm property, we can bring down the exponent t:

t * ln(0.9875) = ln(0.4).

Dividing both sides by ln(0.9875), we obtain:

t = ln(0.4) / ln(0.9875).

Using a calculator, we find that t ≈ 67.6.

Therefore, the sample will have a mass of 20mg approximately 67.6 days after it was set aside for observation.

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X=49°

Step-by-step explanation:

we have an isosceles triangle

thus our equation is

x+x+82=180°

=>2x+82°=180°

cancel 82 from both sides

=>2x=98

divide both sides by 2

therefore

X=49

Answer:

$1400

Step-by-step explanation:

Given that:

80% of the college tuition was paid from the income of Chana.

Next year, Amount to be contributed toward the tuition = $2000

Next year, the tuition will be $600 more than this year's tuition.

To find:

Amount of this year's tuition = ?

Solution:

Let the amount of this year's tuition = $\(x\)

As per the question statement, next year's tuition will be $2000 which is $600 more than this year's tuition fee.

Writing the equation:

\(x+\$600 = \$2000\\\Rightarrow \bold{x = \$1400}\)

This year's tuition fee is $1400.

The solution to the equation 3e^(2x) + 5 = 27 is x = 1.36.

To solve the equation 3e^(2x) + 5 = 27 using natural logarithms, we can follow these steps:

Step 1: Subtract 5 from both sides of the equation:
3e^(2x) = 22

Step 2: Divide both sides of the equation by 3:
e^(2x) = 22/3

Step 3: Take the natural logarithm (ln) of both sides of the equation:
ln(e^(2x)) = ln(22/3)

Step 4: Apply the property of logarithms that states ln(e^a) = a:
2x = ln(22/3)

Step 5: Divide both sides of the equation by 2:
x = ln(22/3)/2

Using a calculator, we can evaluate ln(22/3) to be approximately 2.72.

Therefore, x = 2.72/2 = 1.36.

So, the solution to the equation 3e^(2x) + 5 = 27 is x = 1.36.

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Answer:

see explanation

Step-by-step explanation:

By the Factor theorem, if (x - a) is a factor of f(x) then f(a) = 0

if (r - 4) is a factor then substituting r = 4 into the polynomial should result in it having a value of zero, that is

6\(r^{4}\) - 17r³ - 46r² + 77r - 20 ( substitute r = 4 )

= 6\((4)^{4}\) - 17(4)³ - 46(4)² + 77(4) - 20

= 6(256) - 17(64) - 46(16) + 308 - 20

= 1536 - 1088 - 736 + 288

= 0

Then (r - 4) is a factor of the polynomial

Answer:

Step-by-step explanation:

Answer:

=9

Step-by-step explanation:

3x9=27

27/3=9

The value of x for the given rectangle does not exist and the area of rectangle is 650x-2\(x^{2}\) square meters.

According to the question,

We have the following information:

A farmer wants to fence the rectangular plot which has river on one of its side.

Total fencing = 650 meters

Length of the rectangle = (650-2x) meters

Width of the rectangle = x meters

Now, the total wire used in fencing will be equal to the sum of length and 2 times width.

So, we have the following expression:

650-2x+x+x = 650

650-2x+2x = 650

Now, there will be no variable left in this equation. So, the value of x or width does not exist for this rectangle.

Area of rectangle = length*width

Area = (650-2x)x

Area = 650x-2\(x^{2}\) square meters

Hence, the value of x for the given rectangle does not exist and the area of rectangle is 650x-2\(x^{2}\) square meters.

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Answer:

d

Step-by-step explanation:

shades 2 out of 5 . 2/5 equals 0.40 shades 40% of 100.

Answer:

Step-by-step explanation:

jjj

Answer:

y = 4x^2 + 32x + 28

Step-by-step explanation:

Before we can find the standard form of the quadratic function with the given coordinates, we must first start with the intercept form, whose general equation is

y = a(x - p)(x - q), where

Step 1:  We can plug in (-6, -20) for x and y, -7 for p and -1 for q into the intercept form.  This will allows us to solve for a:

-20 = a(-6 - (-7))(-6 - (-1))

-20 = a(-6 + 7)(-6 + 1)

-20 = a(1)(-5)

-20 = -5a

4 = a

Thus, the full equation in vertex form is

y = 4(x + 7)(x + 1).

Step 1:  The general equation for standard form is

y = ax^2 + bx + c.

We can convert from vertex to standard form by simply expanding the expression.  Let's ignore the 4 for a moment simply focus on (x + 7)(x + 1).

We can expand using the FOIL method, where you multiply

We see that the first terms are x and x, the outer terms are x and 1, the inner terms are 7 and x and the last terms are 7 * 1.  Now, we multiply and simplify:

(x * x) + (x * 1) + (7 * x) + (7 * 1)

x^2 + x + 7x + 7

x^2 + 8x + 7

Step 3:  Now, we can distribute the four to each term with multiplication:

4(x^2 + 8x + 7)

4x^2 + 32x + 28

Optional Step 4:  We can check that our quadratic function in standard form, by plugging in -7, -1, and -6 for x and seeing that we get 0 as the y value for both x = -7 and x = -1 and -20 as the y value for x = -6:

Checking that (-7, 0) lies on the parabola of 4x^2 + 32x + 28:

0 = 4(-7)^2 + 32(-7) + 28

0 = 4(49) - 224 + 28

0 = 196 - 196

0 = 0

Checking that (-1, 0) lies on the parabola of 4x^2 + 32x + 28:

0 = 4(-1)^2 + 32(-1) + 28

0 = 4(1) - 32 + 28

0 = 4 - 4

0 = 0

Checking that (-6, -20) lies on the parabola of 4x^2 + 32x + 28:

-20 = 4(-6)^2 + 32(-6) + 28

-20 = 4(36) -192 + 28

-20 = 144 -164

-20 = -20

I attached a graph from Desmos to show how the function y = 4x^2 + 32x + 28 contains the points (-7, 0), (-1, 0), (-6, 20), further proving that we've correctly found the quadratic function in standard form passing through these three points

The first-order optimality condition for convex optimization problems can be applied to characterize the optimal point, x* in the given optimization problem.

The first-order optimality condition states that if x* is an optimal point for the given convex optimization problem, then there exists a vector v* such that:

∇f(x*) + v* = 0

Here, ∇f(x*) is the gradient of the objective function f(x) evaluated at x*, and v* is the Lagrange multiplier associated with the constraint x ∈ C.

In the given optimization problem, the objective function is ∥a−x∥², and the constraint set is C.

To apply the first-order optimality condition, we need to find the gradient of the objective function. The gradient of ∥a−x∥² is given by:

∇f(x) = 2(x - a)

Now, let's apply the first-order optimality condition to the given problem:

∇f(x*) + v* = 0

Substituting the gradient expression:

2(x* - a) + v* = 0

Rearranging the equation:

x* = a - (v*/2)

This equation provides a characterization of the optimal point x* in terms of the Lagrange multiplier v*. By solving the equation, we can find the optimal point x*.

It's important to note that the Lagrange multiplier v* depends on the constraint set C. The specific form of v* will vary depending on the nature of the constraint set. In some cases, it may be necessary to further analyze the specific properties of the constraint set C to fully characterize the optimal point x*.

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The correct ordering, assuming x is positive, is III: 2x < x² < 2 < 1/x²< 1.

Let's evaluate each option one by one:

I. x² < 2x < 1/x² < 2 < 1

If x is positive, x² will always be greater than 1/x². Therefore, this ordering is not possible.

II. x² < 1/x² < 2x < 1 < 2

Similarly, x² will always be greater than 1/x². Therefore, this ordering is also not possible.

III. 2x < x² < 2 < 1/x² < 1

For this ordering to be true, we need to confirm that 2x is indeed less than x². Since x is positive, we can divide both sides of the inequality by x to preserve the inequality direction. This gives us 2 < x. As long as x is greater than 2, this ordering holds true. Therefore, the correct ordering, assuming x is positive, is III: 2x < x² < 2 < 1/x²< 1.

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Rounding to 2 decimal places, the missing side of the right triangle is approximately 15.62 units.

To find the missing side of a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, we are given sides a = 10 and b = 12, and we need to find the missing side c.

Using the Pythagorean theorem:

c^2 = a^2 + b^2

Substituting the given values:

c^2 = 10^2 + 12^2

c^2 = 100 + 144

c^2 = 244

Taking the square root of both sides to solve for c:

c = √244

c ≈ 15.62

Rounding to 2 decimal places, the missing side of the right triangle is approximately 15.62 units.

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The tangent line to the graph of f is horizontal at x = 0, x = 1, and x = -1.

To find the derivatives and the slope of the graph of f at x = -4, we use the following:

(A) To find f'(x), we take the derivative of f(x):

f(x) = 2x^4 - 4x^2 + 9

f'(x) = 8x^3 - 8x

(B) The slope of the graph of f at x=-4 is given by f'(-4).

f'(-4) = 8(-4)^3 - 8(-4) = -1024

Therefore, the slope of the graph of f at x = -4 is -1024.

(C) The equation of the tangent line to the graph of f at x = -4 can be found using the point-slope form:

y - f(-4) = f'(-4)(x - (-4))

y - f(-4) = f'(-4)(x + 4)

Substituting f(-4) = 2(-4)^4 - 4(-4)^2 + 9 = 321 into the above equation, we get:

y - 321 = -1024(x + 4)

Simplifying, we get:

y = -1024x - 4063

Therefore, the equation of the tangent line to the graph of f at x = -4 is y = -1024x - 4063.

(D) The tangent line is horizontal when its slope is zero. Therefore, we set f'(x) = 0 and solve for x:

f'(x) = 8x^3 - 8x = 0

Factorizing, we get:

8x(x^2 - 1) = 0

This gives us three solutions: x = 0, x = 1, and x = -1.

Therefore, the tangent line to the graph of f is horizontal at x = 0, x = 1, and x = -1.

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The torque is shared between these two materials.

The shear stress in the aluminum rod is obtained asτ_al \(= [(T x 10⁶) / (2.654 x 10⁷)] x [(D_t + D_al)/4]τ_al = (T/663.5) x (60/4)τ_al = (T/44.23) MPa\)

The torque is resisted by both the steel tube and the aluminum rod.

Maximum shear stress in each material,τ_max = (T/J) x (D/2) ,

where D is the diameter of the steel tube or the aluminum rodSteel tube:

The torque is resisted by the steel tube only.

Therefore,τ_max(tube)\(= (T/J) x (D_t/2)τ_max(tube) = [(T x 10⁶) / (2.654 x 10⁷)] x (40/2)τ_max(tube) = (T/663.5) MPa Aluminum rod:\)

Maximum and minimum shear stress in each material are:

Maximum shear stress in steel tube, τ_max(tube) = (T/663.5) MPa

Minimum shear stress in steel tube, τ_min(tube) = -τ_max(tube)

Minimum shear stress in aluminum rod, τ_min(al) = -τ_al

Maximum shear stress in aluminum rod, τ_max(al) = τ_al

The maximum and minimum shear stress in each material can be represented graphically as shown below:

Graphical representation of maximum and minimum shear stress in each material

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The solution to the system of Inequalities is gotten as;  (1, 1)

The system of inequalities given for which we are to graph are;

y ≤ x + 2

y < -2x - 1

Now, to find the solution of the system of Inequalities, we will graph both of them on the same graph and the solution will be the point at which both lines intersect each other.

The graph attached shows that the solution of the inequalities is (1, 1)

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The approximate probability that the baggage limit will not be exceeded on the particular airplane is approximately 0.033, or 3.3%.

To calculate this probability, we need to use the concept of the standard normal distribution. We can convert the given mean and standard deviation of the individual passengers' checked-in baggage weight into a standard normal distribution by applying the formula:

Z = (X - μ) / σ

where Z is the standard score, X is the individual baggage weight, μ is the mean weight, and σ is the standard deviation.

In this case, the maximum allowable weight of the checked baggage for the airplane is 6,000 pounds, and the capacity of the airplane is 125 passengers. So the maximum allowable weight per passenger is 6,000 / 125 = 48 pounds.

Now, we need to find the probability that the baggage weight of a randomly selected passenger is less than or equal to 48 pounds. We can convert this value into a standard score by substituting the values into the formula:

Z = (48 - 42) / 25 = 0.24

We can then look up the probability associated with this standard score in the standard normal distribution table or use a statistical calculator to find that the probability is approximately 0.590.

Since there are 125 passengers on the plane, we need to calculate the probability that all of them have baggage weights less than or equal to 48 pounds. This is done by raising the individual probability to the power of the number of passengers:

P = 0.590^125 ≈ 0.033

Therefore, the approximate probability that the baggage limit will not be exceeded on the particular airplane is approximately 0.033, or 3.3%.

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Answer:

1        -       -3

2       -        9

3       -       -27

4       -        81

5       -       -243

Answer:

okay thnx

Step-by-step explanation:

Answer:

a=36

Step-by-step explanation:

18=a/2

×2 ×2

36=a

Imao thanks 4 the free 5 points

Have a good day bro cya

False. If two nonzero vectors point in the same direction, their dot product will not be zero.

The dot product of two vectors is a mathematical operation that measures the degree of similarity or alignment between the vectors. It is calculated by multiplying the corresponding components of the vectors and summing the results.

When two vectors point in the same direction, their dot product will be positive, indicating that they are aligned or have a certain degree of similarity. The dot product will be equal to the product of the magnitudes of the vectors multiplied by the cosine of the angle between them.

If two nonzero vectors are parallel and point in the same direction, the angle between them is 0 degrees, and the cosine of 0 degrees is 1. Therefore, the dot product of two nonzero vectors pointing in the same direction will be equal to the product of their magnitudes.

In other words, if the vectors are represented as A and B, and they point in the same direction, the dot product (A · B) will be equal to ||A|| * ||B||, where ||A|| and ||B|| represent the magnitudes of the vectors A and B, respectively.

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For a function, f(x) = 1 + 3x , the computed value of lim h→0 [f(5 + h) − f(5)]/h is equals to the three.

In Mathematics, the limit of a function is a value of the function as the input of the function tries approaches some number. Function limits are used to define continuity, integrals, and derivatives. Mathematically, let f(x) be a function and L be limit of function, f(x) at x a exist if and only if lim f(x) = lim f(x) = L

x→a⁻ x→a⁺

We have, f(x) = 1+3x and will compute lim h→0 [f(5 + h) − f(5)]/h . Firstly, determine the value of function at x = 5 , x = 5+h.

So, f(5) = 1+ 3×5 = 16 and f(5+ h) = 1+3(5+h)

= 16+ 3h

Now, lim [ f(5 + h) − f(5)]/ h

h→0

= lim [(16 + 3h) − (16) ]/ h

h→0

= lim [ 16 + 3h - 16 ]/ h

h→0

= lim [ 3h/ h ] = lim [ 3h¹h⁻¹ ]

h→0 h→0

= lim [ 3h¹⁻¹ ] = lim [ 3h⁰ ]

h→0 h→0

= lim 3 (1) = 3 ( since, x⁰ = 1 )

h→0

Hence, the required limit value is 3.

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Answer:

There are infinitely many solutions to the system because the equations represent the same line

Answer:

75%

Step-by-step explanation:

\(\frac{15}{20}\) as a percent = 75%, or 0.75 in decimal form.

We can easily see this by noting that if we multiply this matrix with \underset{x}{\rightarrow}= (x1,x2,x3), we get

|x1 + 2x2 + 3x3|

|4x1 + 5x2 + 6x3|

|7x1 + 8x2 + 9x3|

which to the equation 3x1 + 2x2 + x3 = 0.

A 3x3 matrix for which the kernel is the set of all vectors \underset{x}{\rightarrow}∈ R3 satisfying 3x1 + 2x2 + x3 = 0 is

|1  2  3|

|4  5  6|

|7  8  9|

This matrix has a kernel consisting of all vectors \underset{x}{\rightarrow}= (x1,x2,x3) that satisfy the equation 3x1 + 2x2 + x3 = 0.

We can easily see this by noting that if we multiply this matrix with \underset{x}{\rightarrow}= (x1,x2,x3), we get

|x1 + 2x2 + 3x3|

|4x1 + 5x2 + 6x3|

|7x1 + 8x2 + 9x3|

which to the equation 3x1 + 2x2 + x3 = 0.

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