Compare and contrast equation notation and function notation

15 + 1.5x = 255

and 160 laps

Answer:

[tex](x+1)[/tex] and [tex](x-4)[/tex].

Step-by-step explanation:

We have been a trinomial [tex]x^2-3x-4[/tex]. We are asked to find the factors of our given trinomial.

We will use split the middle term to solve our given problem. We need to two number whose sum is [tex]-3[/tex] and whose product is [tex]-4[/tex]. We know such two numbers are [tex]-4\text{ and }1[/tex].

Split the middle term:

[tex]x^2-4x+x-4[/tex]

Make two groups:

[tex](x^2-4x)+(x-4)[/tex]

Factor out GCF of each group:

[tex]x(x-4)+1(x-4)[/tex]

[tex](x-4)(x+1)[/tex]

Therefore, factors of our given trinomial are [tex](x+1)[/tex] and [tex](x-4)[/tex].

15 ft = 15*12 = 180 inches

180/4 = 45

 so the scale is 1/45

The only correct statement among the choices is:

A. When a pair of parallel lines are intersected by a third line, the alternate interior angles are congruent.

This statement can actually be proven by measuring the angles of the alternate interior angles. We can actually see that they have equal measurements. While the adjacent interior angles are supplementary angles.

Answer:15 2 3

Step-by-step explanation:

The partial derivatives fx(30,5) and ft(30,5) can be calculated using the given information, and the tangent plane approximation can be used to estimate f(33,4).

The chemical reaction time is represented by the function f(t,x), where t is the temperature in degrees Celsius and x is the quantity of a catalyst present in grams. We are given that when the temperature is 30 degrees Celsius and there are 5 grams of catalyst, the reaction takes 50 minutes. Additionally, increasing the temperature by 3 degrees reduces the time taken by 5 minutes, and increasing the amount of catalyst by 2 grams decreases the time taken by 3 minutes.

To find the partial derivative fx(30,5), we need to find the rate of change of the reaction time with respect to the quantity of catalyst, while holding the temperature constant. Using the given information, we can calculate:

fx(30,5) = (f(30,5+2)-f(30,5))/2 = (-3)/2 = -1.5

To find the partial derivative ft(30,5), we need to find the rate of change of the reaction time with respect to the temperature, while holding the quantity of catalyst constant. Using the given information, we can calculate:

ft(30,5) = (f(30+3,5)-f(30,5))/3 = (-5)/3 = -1.67

Using the tangent plane approximation, we can estimate f(33,4) by calculating the change in reaction time by adjusting the temperature to 33 degrees Celsius and the quantity of catalyst to 4 grams.

f(33,4) = f(30,5) + ft(30,5) * (33-30) + fx(30,5) * (4-5)

f(33, 4) = 50 + (-1.67) * 3 + (-1.5) * -1 = 50 - 5.01 + 1.5 = 46.49

[tex]\( f_t(30,5) = -\frac{5}{3} \)[/tex], [tex]\( f_x(30,5) = -1.5 \)[/tex], [tex]\( f(33,4) \approx 46.5 \)[/tex] minutes.

ft (30, 5) =_[tex]\(-\frac{5}{3} \)[/tex]_ fx (30,5) =_-1.5_ f (33,4)=__46.5 minutes__

To find the partial derivatives [tex]\( f_t(30,5) \)[/tex] and [tex]\( f_x(30,5) \)[/tex], we'll use the given information about how changes in [tex]\( t \) and \( x \)[/tex] affect the time [tex]\( f(t,x) \)[/tex].

Given:

- [tex]\( f(30,5) = 50 \)[/tex] minutes

- Increasing [tex]\( t \)[/tex] by 3 degrees Celsius reduces [tex]\( f \)[/tex] by 5 minutes.

- Increasing [tex]\( x \)[/tex] by 2 grams decreases [tex]\( f \)[/tex] by 3 minutes.

1. Partial derivative with respect to [tex]\( t \) at \( (30,5) \)[/tex]:

  - Change in [tex]\( t \): \( \Delta t = 3 \)[/tex] degrees Celsius

  - Change in [tex]\( f \): \( \Delta f = -5 \)[/tex] minutes

  - Using the definition of partial derivative:

    [tex]\[ f_t(30,5) = \frac{\Delta f}{\Delta t} = \frac{-5}{3} = -\frac{5}{3} \text{ minutes/degree Celsius} \][/tex]

2. Partial derivative with respect to [tex]\( x \) at \( (30,5) \)[/tex]:

  - Change in [tex]\( x \): \( \Delta x = 2 \)[/tex] grams

  - Change in [tex]\( f \): \( \Delta f = -3 \)[/tex] minutes

  - Using the definition of partial derivative:

    [tex]\[ f_x(30,5) = \frac{\Delta f}{\Delta x} = \frac{-3}{2} = -1.5 \text{ minutes/gram} \][/tex]

Now, to find [tex]\( f(33,4) \)[/tex] using the tangent plane approximation, we'll use the partial derivatives we found and the point [tex]\( (30,5) \)[/tex] as a reference:

The tangent plane approximation formula is:

[tex]\[ f(t,x) \approx f(30,5) + f_t(30,5)(t-30) + f_x(30,5)(x-5) \][/tex]

Substituting the values:

[tex]\[ f(33,4) \approx 50 + \left(-\frac{5}{3}\right)(33-30) + (-1.5)(4-5) \][/tex]

[tex]\[ f(33,4) \approx 50 - 5 + 1.5 \][/tex]

[tex]\[ f(33,4) \approx 46.5 \text{ minutes} \][/tex]

So, [tex]\( f(33,4) \approx 46.5 \)[/tex] minutes.

Thus:

- [tex]\( f_t(30,5) = -\frac{5}{3} \)[/tex] minutes/degree Celsius

- [tex]\( f_x(30,5) = -1.5 \)[/tex] minutes/gram

- [tex]\( f(33,4) \approx 46.5 \)[/tex] minutes

The correct question is:

The number of minutes taken for a chemical reaction if f(t,x). It depends on the temperature t degrees Celsius, and the quantity, x grams, of a catalyst present. When the temperature is 30 degrees Celsius and there are 5 grams of catalyst, the reaction takes 50 minutes. Increasing the temperature by 3 degrees reduces the time taken by 5 minutes. Increasing the amount of catalyst by 2 grams decreases the time taken by 3 minutes. Use this information to find the partial derivatives fx(30,5) and ft(30,5). Use the tangent plane approximation to find f(33,4). ft (30, 5) =__________ fx (30,5) =__________ f (33,4)=__________

Final answer:

By dividing the total length of the wire (36 cm) by the number of sides of a square (4), we find that the length of one side of the square is 9 cm.

Explanation:

We start by calculating the total length of the wire when it was bent into an equilateral triangle. Since each side is 12 cm long and there are three sides, the total wire length is 3 × 12 cm = 36 cm. When the wire is unbent and reshaped into a square, the total wire length remains the same; thus, the perimeter of the square is also 36 cm. A square has four equal sides, so the length of each side of the square is the perimeter divided by four.

Length of one side of the square = Total wire length ÷ 4

Length of one side of the square = 36 cm ÷ 4

Length of one side of the square = 9 cm

The required numbers are "15" and "2".

A number system is defined as a way to represent numbers on the number line using a set of symbols and approaches. These symbols, which are known as digits, are numbered 0 through 9.

Let the first number would be x and the second number would be y

We have been given that the two numbers with a product of 30,

So, x × y = 30

x = 1/y   .....(i)

And the sum of 17, then x + y = 17   .....(ii)

Take both equations, then solves them and we get the values that are:

x = 15 and y = 2

So, 15 × 2 = 30 and 15 + 2 = 17

Thus, the required numbers are "15" and "2".

Learn more about the number system here:

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Answer:

Approximately 8.6

Step-by-step explanation:

Answer:  The required value of f(x) for x = 1 is 11.

Step-by-step explanation:  We are given to evaluate f(x) for x = 1, where

[tex]f(x)=3x+8~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

To evaluate the required value, we need to substitute x = 1 in expression (i).

So, putting x = 1 in (i), we get

[tex]f(1)=3\times1+8=3+8=11.[/tex]

Thus, the required value of f(x) for x = 1 is 11.

The cost of one can of tomatoes is $1.30, so nine cans at this rate would cost $11.70.

If five cans of tomatoes cost $6.50, to find the cost of nine cans, we first need to find the cost of one can of tomatoes. We do this by dividing the total cost by the number of cans.

Cost per can = Total cost / Number of cans

Cost per can = $6.50 / 5

Cost per can = $1.30

Now that we know how much one can costs, we can find out how much nine cans will cost by multiplying the cost per can by nine.

Total cost for nine cans = Cost per can × Number of cans

Total cost for nine cans = $1.30 × 9

Total cost for nine cans = $11.70

Therefore, nine cans of tomatoes will cost $11.70.

The epicenter is located at (-5, 1) on the coordinate plane.

To find the coordinates of the epicenter, we need to find the average of the coordinates of the three receiving stations. The x-coordinate of the epicenter is (3 - 13 - 5)/3 = -5, and the y-coordinate is (3 - 5 + 3)/3 = 1. Therefore, the epicenter is located at (-5, 1) on the coordinate plane.

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The coordinates of the epicenter of earthquake are (-1, 0).

The epicenter of the earthquake is the point that is equidistant from stations X, Y, and Z. This is essentially the solution to a system of three equations representing the distances from the epicenter to each of the three stations.

The equations are derived from the distance formula, [tex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex] ,

where [tex](x_1,y_1)[/tex] and [tex](x_2, y_2)[/tex] are the coordinates of the two points and d is the distance between them.

For station X at (3,3), the distance to the epicenter (x,y) is 5 units. So, the equation is:

[tex](x - 3)^2 + (y - 3)^2 = 5^2 = 25 \text{ ....(1)}[/tex]

For station Y at (-13,-5), the distance to the epicenter is 13 units. So, the equation is:

[tex](x + 13)^2 + (y + 5)^2 = 13^2 = 169\text{ ....(2)}[/tex]

For station Z at (-5,3), the distance to the epicenter is 5 units. So, the equation is:

[tex](x + 5)^2 + (y - 3)^2 = 5^2=25 \text{ ....(3)}[/tex]

Solving this system of equations will give the coordinates of the epicenter.

Subtract equation (1) from equation (2), we get:

[tex](x^2+26x+169+y^2+10y+25)-(x^2-6x+9+y^2-6y+9) = 144\\\\\text{Simplifying, we get:}\\32x+16y+176=144\\32x+16y = -32\\\text{Divide by 16 on both sides of the equation, we get:}\\2x + y = -2[/tex]

Subtract equation (1) from equation (3), we get:

[tex](x+5)^2+(y-3)^2-(x-3)^2-(y-3)^2 = 0\\x^2+10x+25-(x^2-6x+9)=0\\16x + 16 = 0\\\text{Dividing by 1 on both sides, we get}\\x + 1 =0\\\text{We get:}\\x = -1[/tex]

Putting x = -1 in the equation 2x + y = -2, we get:

2(-1) + y = -2

We get,

y = 0

So, the coordinates of the epicenter of earthquake are (-1, 0).

The probability of choosing a remote control car from the box is 1/3, which is calculated by dividing the number of remote control cars (5) by the total number of toys (15).

To find the probability of choosing a control car from the box, we use the formula for probability, which is:

Probability = Number of favorable outcomes / Total number of possible outcomes

The number of favorable outcomes is the quantity of remote control cars which is 5. The total number of toys is the sum of all the toys, which are 4 Lego sets + 6 video games + 5 remote control cars = 15 toys in total.

The probability is therefore calculated as:

Probability of choosing a remote control car = 5 (number of remote control cars) / 15 (total number of toys) = 1/3

Hence, the correct answer is C. 1/3.