What is the constant of proportionality for the ratios 2 to 3 and 4 to 6?

Final answer:

To model the situation, use the rule t = 12 + 8(n - 1), where n is the number of trucks. Applying this rule, 7 trucks can carry a total of 60 tons of dirt.

Explanation:

The student should write a rule to model the situation where the first truck carries 12 tons of dirt and each subsequent truck carries 8 tons. The rule can be defined as t = 12 + 8(n - 1), where n represents the number of trucks and t represents the total tons of dirt carried. For 7 trucks, we can plug in the value of n into the equation:

t = 12 + 8(7 - 1)

t = 12 + 8(6)

t = 12 + 48

t = 60 tons

Therefore, 7 trucks can carry a total of 60 tons of dirt.

The slope-intercept form y = mx + b shows how the slope and y-intercept values are used in the equation, while the point-slope form y - y₁ = m(x - x₁) defines a line using a point and the slope.

The slope-intercept form, y = mx + b, makes sense as a name because it explicitly shows how the slope and y-intercept values are used in the equation.

The 'm' represents the slope, which describes the steepness of the line, and the 'b' represents the y-intercept, which indicates where the line intersects the y-axis.

For example, in the equation y = 2x + 3, the slope is 2 and the y-intercept is 3.

The point-slope form, y - y₁ = m(x - x₁), is called so because it defines a line using a single point (x₁, y₁) and the slope 'm'. This form makes it easier to determine the equation of a line when given a point and the slope.

For instance, in the equation y - 2 = -3(x - 4), the point (4, 2) and the slope -3 are used to specify the line equation.

Final answer:

To find the dimensions of the picture, set up an equation using the length and width. Solve the equation to find the dimensions: length = 32 inches, width = 15 inches.

Explanation:

To find the dimensions of the picture, let's assume the length is x inches. According to the problem, the width is 17 inches shorter than the length, so the width would be (x - 17) inches. The total area of the picture is given as 480 square inches. We can set up an equation using the length and width: x(x - 17) = 480. Solve this quadratic equation to find the dimensions of the picture.

Expanding the equation, we get x^2 - 17x - 480 = 0. Factoring the quadratic equation, we find (x + 15)(x - 32) = 0. Therefore, the possible values for x are -15 and 32. Since the length cannot be negative, we discard -15. Thus, the length of the picture is 32 inches and the width is (32 - 17) = 15 inches.

Final answer:

The dimensions of the picture are 32 inches in length and 15 inches in width.

Explanation:

To find the dimensions of the picture, we need to establish equations based on the information provided. Let the length of the picture be L inches and the width be W inches.

Given that:

Plug the width expression from step 1 into the area equation from step 2 to form a quadratic equation:

L(L - 17) = 480

Solving this quadratic equation will give us the value of L. Once we have L, we can use the first equation to get W. Let's solve the quadratic equation:

L² - 17L - 480 = 0

Factoring the quadratic, we find that (L - 32)(L + 15) = 0. So L can be either 32 or -15. Since a length cannot be negative, L must be 32 inches, and W, therefore, is 32 - 17 = 15 inches.

The dimensions of the picture are therefore 32 inches by 15 inches.

Farmer Jack needs an additional 1,602 2/3 square feet of land to plant the rest of his corn.

To find out how much more land Farmer Jack needs to plant the rest of his corn, we need to calculate the area of his current garden and subtract it from the required area of 1,800 square feet. Farmer Jack's garden is 10 5/6 feet by 18 feet, so we can multiply these two dimensions to get the area of his current garden. Then, we subtract this area from 1,800 to find out how much more land he needs:

Area of Farmer Jack's garden: 10 5/6 feet * 18 feet = 197 1/3 square feet

More land needed by Farmer Jack: 1,800 square feet - 197 1/3 square feet = 1,602 2/3 square feet

Farm Jack needs an additional 1,602 2/3 square feet of land to plant the rest of his corn.

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Final answer:

The derivative dy/dx for the equation 9y cos(x) = x² + y² is found by applying implicit differentiation. Rearranging terms after differentiation, dy/dx is obtained as the fraction (9y sin(x) + 2x) / (9 cos(x) - 2y).

Explanation:

The question asks for the derivative of dy/dx using the equation 9y cos(x) = x2 + y2. To find dy/dx, we can apply implicit differentiation, which implies taking the derivative of both sides with respect to x while treating y as a function of x (y(x)).

Let's differentiate both sides:

The left side: d/dx [9y cos(x)] = d/dx [9y] cos(x) + 9y d/dx[cos(x)] = 9(dy/dx) cos(x) - 9y sin(x)

The right side: d/dx [x2 + y2] = 2x + 2y(dy/dx)

Equating both the derivatives, we have:

9(dy/dx) cos(x) - 9y sin(x) = 2x + 2y(dy/dx)

Now, solving for dy/dx gives us:

dy/dx = (9y sin(x) + 2x) / (9 cos(x) - 2y)

Answer:

when the tension is 72 pounds , the frequency is 900 hertz

Step-by-step explanation:

Hello, I think I can help you with this.

you can easily solve this by using a rule of three.

According to the question data:

the pitch, or frequency, of a vibrating string varies directly with the square root of the tension.in mathematical terms it is:

f ∝ √T

where f is the pitch or frequency and T is the tension

Step 1

if a string vibrates at a frequency of 300 hertz due to a tension of 8 pounds

300 hertz  ∝ √8 pounds

300⇔√8

what is the frequency when the tension is 72 pounds

x⇔√72

Step 2

Let

300⇔√8

x⇔√72

the relation is

[tex]\frac{300}{\sqrt{8} } =\frac{x}{\sqrt{72} }\\\\ Now, solve\ for\ x\\\\\\\frac{300*\sqrt{72} }{\sqrt{8} } =x\\x=\frac{300*\sqrt{72} }{\sqrt{8} } \\x=\frac{300*\sqrt{8*9}}{\sqrt{8}} \\x=\frac{300*\sqrt{8}*\sqrt{9}}{\sqrt{8}} \\x=300*\sqrt{9}\\x=300*3\\x=900[/tex]

when the tension is 72 pounds , the frequency is 900 hertz

have a good day.

To deliver 2 grams of a drug with a concentration of 350 mg per 10 mL, convert 2 grams to milligrams (2000 mg), and then divide this value by the concentration in mg/mL (350 mg/10 mL) to find the volume in milliliters. After calculation, this results in approximately 6 milliliters, when rounded to the nearest milliliter.

To calculate how many milliliters are needed to deliver 2 grams of the drug when the concentration is 350 mg per 10 mL, we start with the conversion of grams to milligrams since the concentration is provided in milligrams. Remember, 1 gram equals 1000 milligrams:

Therefore, approximately 6 milliliters of the drug are needed to deliver a dose of 2 grams.

Each bag contained 1.375 cups of sugar.

To find out how many cups of sugar were in each bag that Latrell bought, we need to divide the total amount of sugar by the number of bags. Latrell has a total of 5 1/2 cups of sugar, which is the same as 5.5 cups when converted to a decimal. He bought 4 bags of sugar. Therefore, the calculation we need to perform is:

5.5 cups / 4 bags = 1.375 cups per bag

So, each bag contained 1.375 cups of sugar.

The solution is : n = 9

In mathematics, multiplication is a method of finding the product of two or more numbers. It is one of the basic arithmetic operations, that we use in everyday life.

Explanation:

To find the answer we need to check at what point the running time of algorithm A is equal the running time of B:

1000*n*n = n*n*n*n*n  

1000 = n*n*n = n^3

n = ∛(1000) = 10  (when n equals ten A equal B)

For any integer greater than ten B > A  and for any integer smaller than ten B < A.

The greatest integer for which B < A is nine, therefore nine is our answer.  

                   A                                      B

                   1000n^2                          n^5

n = 8             64000              <             32768

n = 9             81000              <             59049

n = 10            100000            =             100000

n = 11             121000             >             161051

n = 12            144000             >             248832

therefore nine is our answer.  

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For [tex]\( n = 0, 1, 2, 3, 4, 5, 6 \)[/tex], Algorithm A (which runs in [tex]\( 2n \)[/tex] steps) beats Algorithm B (which runs in [tex]\( 5\sqrt{n} \)[/tex] steps).

To determine for which values of [tex]\( n \)[/tex] Algorithm A beats Algorithm B in terms of running time, we compare their complexities:

Algorithm A runs in [tex]\( 2n \)[/tex] steps.

Algorithm B runs in [tex]\( 5\sqrt{n} \)[/tex] steps.

Algorithm A beats Algorithm B if [tex]\( 2n < 5\sqrt{n} \)[/tex].

Let's solve this inequality step by step:

1. Square both sides to eliminate the square root (valid since [tex]\( n \geq 0 \)[/tex]):

[tex]\[ (2n)^2 < (5\sqrt{n})^2 \][/tex]

[tex]\[ 4n^2 < 25n \][/tex]

2. Bring all terms to one side to form a quadratic inequality:

[tex]\[ 4n^2 - 25n < 0 \][/tex]

3. Factorize the quadratic inequality if possible:

[tex]\[ n(4n - 25) < 0 \][/tex]

4. Find the values of [tex]\( n \)[/tex] that satisfy the inequality:

[tex]\( n(4n - 25) < 0 \)[/tex] holds when one factor is negative, and the other is positive:

[tex]\( n < 0 \)[/tex] and [tex]\( 4n - 25 > 0 \)[/tex]: No valid solutions since [tex]\( n \geq 0 \)[/tex].

[tex]\( n > 0 \)[/tex] and [tex]\( 4n - 25 < 0 \)[/tex]:

[tex]\[ 4n < 25 \Rightarrow n < \frac{25}{4} = 6.25 \][/tex]

[tex]\( n < 0 \) and \( 4n - 25 < 0 \)[/tex]: All [tex]\( n \geq 0 \)[/tex] satisfy this condition.

5. Determine the integer values of [tex]\( n \)[/tex]:

Since [tex]\( n \)[/tex] must be non-negative, the integer values of [tex]\( n \)[/tex] that satisfy [tex]\( 4n^2 - 25n < 0 \)[/tex] are [tex]\( n = 0, 1, 2, 3, 4, 5, 6 \)[/tex].

The answer is; c.) checks!

The equation of parabola is y = ( x - 2 )² + 3

What is a Parabola?

A Parabola, open curve, a conic section produced by the intersection of a right circular cone and a plane parallel to an element of the cone. A parabola is a plane curve generated by a point moving so that its distance from a fixed point is equal to its distance from a fixed line

The equation of the parabola is given by

( x - h )² = 4p ( y - k )

where ( h , k ) is the vertex and ( h , k + p ) is the focus

y is the directrix and y = k - p

The equation of the parabola is also given by the equation

y = ax² + bx + c

where a , b , and c are the three coefficients and the parabola is uniquely identified

Given data ,

The vertex of the parabola is given as A ( 2 , 3 )

Now ,

The equation of parabola is given by

y = a ( x - h )² + k

where the vertex of the parabola is A ( h , k )

a = 1

So , substituting the values of vertex in the equation of parabola , we get

y = ( x - 2 )² + 3

Hence , the equation of parabola is y = ( x - 2 )² + 3

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Answer:14,328

Step-by-step explanation:

To convert [tex]7.085 * 10^-{14}[/tex] to an ordinary number, move the decimal point 14 places to the left, resulting in 0.00000000000007085.

To write [tex]7.085 * 10^-{14}[/tex] as an ordinary number, you need to move the decimal point 14 places to the left because the exponent is negative. This means the number is very small. The process is similar to changing other numbers from scientific to standard notation, for example, changing [tex]7.5 x 10^{-3}[/tex] to 0.0075 by moving the decimal point three places to the left.

For [tex]7.085 * 10^-{14}[/tex], you will end up with 0.00000000000007085. The number has thirteen zeros after the decimal point and before the 7085 because we move the decimal 14 places.

Answer:

the Answer is. A 1/7

Step-by-step explanation:

i did the test