Steve puts only dimes and quarters into his piggy bank. Right now he has five more dimes than quarters there, and they make $74.35. How many quarters and how many dimes are there in Steve’s piggy bank?

The rate of change and the initial value are given as -5 and 27 respectively.

To represent a straight line on a graph consider two points namely x and y intercepts of the line. To find x-intercept put y = 0 and for y-intercept put x = 0. Then draw a line passing through these two points.

Given that,

The points for the given linear function are given as (0, 27) and (5, -8) respectively.

The rate of change is equivalent to the slope of the straight line and is given as,

(-8 - 27)/(5 - 0) = -35/7 = -5

Now, the equation of the function can be written as,

(y - 27)/(x - 0) = -5

=> y - 27 = -5x

=> y = -5x + 27

The initial value is given at x = 0 as below,

y = -5 × 0 + 27

  = 27

Hence, the rate of change for the function is -5 and its initial value is 27.

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Final answer:

To factor the expression 10x-25y, we find the greatest common factor, which is 5, and write the expression as 5(2x - 5y).

Explanation:

To factor the expression 10x−25y, we need to find the greatest common factor (GCF) that both terms share. In this case, both terms are divisible by 5. When we factor out the GCF, we get:

Identify the GCF of 10x and 25y, which is 5.

Divide both terms by the GCF: 10x/5 = 2x and 25y/5 = 5y.

Write the factored expression: 5(2x - 5y).

Thus, the expression 10x−25y factored is 5(2x - 5y)

The required, there are 108 cups of water in the pail as a pail hold 6 3/4 gallon of water.

Unit of measurement is defined as every entity having its measures in dimensions, weight, and time. Such as for length we have a meter, for liquid we have liters, and so on.

Here,
To convert 6 3/4 gallons to cups, we can use the fact that there are 16 cups in one gallon.

So, we can start by multiplying 6 (the number of whole gallons) by 16 to get the number of cups in those gallons:

6 gallons × 16 cups/gallon = 96 cups

Next, we need to convert the 3/4 of a gallon to cups. To do this, we can multiply the fraction by the number of cups in one gallon:

3/4 gallon × 16 cups/gallon = 12 cups

So, the total amount of water in the pail is:

96 cups + 12 cups = 108 cups

Therefore, there are 108 cups of water in the pail.

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The recipe can make serving 8 cups.

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.

Here, given that,

A recipe makes 6 2/3 cups

                         =20/3 cups

And, the serving size is 5/6 cup

So, the recipe can make serving = 20/3÷ 5/6 cups

                                                      =20/3 × 6/5 cups

                                                      =8 cups

Hence, the recipe can make serving 8 cups.

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To measure the approximately 4,000 km long Mississippi River with M&Ms, we will need about 400,000,000 M&Ms, as there are 100 cm in a meter and 1,000 m in a km.

The question involves estimating the length of the Mississippi River using the size of M&Ms as a unit of measurement. Since the river is about 4,000 kilometers long and the standard unit of length in the SI system is the meter, we need to convert kilometers to centimeters to match the size of an M&M, which is about 1 centimeter long. There are 100 centimeters in a meter, and 1,000 meters in a kilometer. Therefore, we have:

Now, since each M&M is 1 centimeter, the number of M&Ms needed to measure the length of the Mississippi River is equal to the total number of centimeters:

400,000,000 M&Ms would be required to measure the approximate length of the Mississippi River.

Final answer:

The snail moved a total distance of 2 1/9 yards after converting 76 inches to yards using the conversion factor of 1 yard equals 36 inches.

Explanation:

To convert the distance a snail moved from inches to yards, we use the conversion factor that 1 yard equals 36 inches. Given that the snail moved 76 inches, we divide this number by 36 to find the distance in yards.

76 inches ÷ 36 inches/yard = 2 yards and 4 inches.

Since there are 36 inches in a yard, the remaining 4 inches must be converted to a fraction of a yard by dividing by 36:

4 inches ÷ 36 = 1/9 yard.

Therefore, the snail moved 2 yards and 1/9 yard, or 2 1/9 yards in total.

The student's question is related to dividing 17 by 4520. This is performed by a straight division resulting in a small decimal quotient, not a product. The calculation is best done using a calculator and rounding to the proper number of significant figures.

The equation given, 17 by 4520, is asking for the result of dividing 17 by 4520. To compute this, we simply perform the division, recognizing that since 4520 is much larger than 17, the answer will be a very small decimal value. The term 'product' in the student's question seems to be a mistake, as they are likely referring to the quotient obtained from the division.

When attempting to perform the computation mentally and simplify the process, one might attempt to find numbers that are more easily divisible or apply scientific notation to simplify calculations, much like re-expressing numbers to make them more manageable. However, for this division, no mental math trick simplifies it effectively, and using a calculator might be the best option.

When going through this process, it is also important to consider significant figures when giving the final answer, rounding as necessary based on the precision of the data provided.

The equivalent expression to [tex]x^2 + 2x - 24 / (3x + 18([/tex] is (x - 4) / 3, after factoring and simplifying the common terms.

The question is asking for an expression equivalent to [tex]x^2 + 2x - 24 / (3x + 18[/tex].

This expression can be factored and simplified.

First, notice that the quadratic expression in the numerator can be factored into (x + 6)(x - 4).

Similarly, the denominator can be factored as 3(x + 6). After factoring, we have:

= (x + 6)(x - 4) / 3(x + 6)

We can now simplify the expression by canceling out the common factor of (x + 6):

= (x + 6)(x - 4) / 3(x + 6) = (x - 4) / 3

So the equivalent expression is (x - 4) / 3.

Answer:

the value of cos θ is [tex]\frac{1}{2}[/tex].

Step-by-step explanation:

Since, [tex] cos\theta=\frac{1}{sec\theta}[/tex]

we are given with sec θ = 2

To find the value of cos θ , we use this fact [tex] cos\theta=\frac{1}{sec\theta}[/tex]

put sec θ = 2 in [tex] cos\theta=\frac{1}{sec\theta}[/tex]

[tex] cos\theta=\frac{1}{2}[/tex]

Therefore, the value of cos θ is [tex]\frac{1}{2}[/tex].

Answer:

Step-by-step explanation:

120 x 5 = 600

The motorcycle is 3 times as fast.

600 x 3 = 1800

Answer:

b

Step-by-step explanation:

Answer:

[tex]\displaystyle \frac{dy}{dt} = \frac{-3}{2(4t- 3)^2\sqrt{\frac{t}{4t - 3}}}[/tex]

General Formulas and Concepts:

Calculus

Differentiation

Derivative Property [Multiplied Constant]:                                                           [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]

Derivative Property [Addition/Subtraction]:                                                         [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]

Basic Power Rule:

Derivative Rule [Quotient Rule]:                                                                           [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]

Derivative Rule [Chain Rule]:                                                                                 [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle y = \sqrt{\frac{t}{4t - 3}}[/tex]

Step 2: Differentiate

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

Answer: We find the equation that represents the temperature (t) at hour (t) using the slope-intercept form and it is:

t = -2 h + 32

Solution:

The temperature, t, in Burrtown starts at 32°F at midnight, when h = 0. This represents the y-interception of the right line, when h=0 hours, t=32 °F:

Point=(h,t)→Point=(0,32)=(0,b)→b=32

For the next few hours, the temperature drops 2 degrees every hour. This represents the slope of the line (m): m=-2, negative because the temperature drops.

Slope-Intercept form:

y=mx+b; y=t, m=-2, x=h, b=32

Replacing the values:

t=-2h+32

Answer:

t=-2h+32

Step-by-step explanation:

Using the normal distribution, it is found that the statement is False, thus the correct option is b.

In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

For this problem, z = 0.3 has a p-value of 0.6554.

1 - 0.6554 = 0.3446

The proportion in the tail beyond z = 0.30 is p = 0.3446, thus, the statement is False.

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if both fractions are less than  1/2 then the sum has to be less than 1

1/8 + 3/8 = 4/8 = 1/2 , less than 1

7/16 + 1/4 = 11/16, less than 1

3/16 + 1/16 = 4/16 = 1/4, less than 1