Find the area of the indicated region. We suggest you graph the curves to check whether one is above the other or whether they cross, and that you use technology to check your answer. HINT [See Example 2.]
Between y = x and y = x2 for x in [−2, 1]

The mathematical symbol that makes the expression -3/4 -11/12 true is subtraction.

The student is likely asking which mathematical symbol (operation) would make the expression -3/4 -11/12 a true sentence. The symbols that could apply are addition (+), subtraction (-), multiplication (x), or division (). To determine which symbol makes the sentence true, one would typically compare the result of applying each operation to the numeric values given.

For example, to test addition:

-3/4 + (-11/12) = (-9/12) + (-11/12) = -20/12 = -5/3

To test subtraction:

-3/4 - (-11/12) = (-9/12) - (-11/12) = 2/12 = 1/6

Therefore, subtraction makes the original expression a true one, because subtracting a negative is the same as adding the positive equivalent, which would simplify to an expression that equals 1/6.

The probability that the second number rolled is greater than or equal to the first number when rolling a fair eight-sided die twice is 1/16.

To find the probability that the second number rolled is greater than or equal to the first number when rolling a fair eight-sided die twice, we need to determine the favorable outcomes and the total number of possible outcomes.

There are 8 possible outcomes for the first roll and 8 possible outcomes for the second roll. However, since the die is fair and has different numbers on each side, the second roll can only be greater than or equal to the first roll for 4 of the outcomes.

Therefore, the probability is 4 favorable outcomes out of 64 total outcomes, which simplifies to a probability of B

The total cost of roasting 110 lb of coffee is $765.

Based on the given function, the marginal cost of roasting x pounds of coffee is:

C'(x) = -0.018x + 4.75 for x ≤ 200.

To find the total cost of roasting 110 lb of coffee, we need to integrate the marginal cost function to get the total cost function C(x), and then evaluate C(110).

However, since we are disregarding any fixed costs, we can assume that the total cost function is simply the integral of the marginal cost function.

Integrating C'(x) = -0.018x + 4.75 with respect to x, we get:

C(x) = -0.009x² + 4.75x + C

where C is the constant of integration. Since we are disregarding any fixed costs, we can assume that C = 0.

Therefore, the total cost of roasting 110 lb of coffee is:

C(110) = -0.009(110)² + 4.75(110) = 765

The total cost of roasting 110 lb of coffee is $765.

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Final answer:

To find the total cost of roasting 110 lb of coffee, integrate the marginal cost function C'(x) = -0.018x + 4.75 over the range of x values. Substitute 110 for x in the total cost function to find the total cost.

Explanation:

To find the total cost of roasting 110 lb of coffee, we need to integrate the marginal cost function over the desired range of x values. The marginal cost function is given by C'(x) = -0.018x + 4.75. Integrating this function will give us the total cost function C(x).

Integrating C'(x) gives C(x) = -0.009x^2 + 4.75x + C, where C is the constant of integration. Since we are disregarding any fixed costs, we can ignore the constant term C.

Now we can substitute 110 for x in the total cost function to find the total cost of roasting 110 lb of coffee: C(110) = -0.009(110)^2 + 4.75(110) = -108.9 + 522.5 = $413.6.

To find the total cost of three types of tickets priced at $8.75, $6.50, and $6.50, add the prices together, resulting in a total cost of $21.75.

To write an expression for the total cost of purchasing three types of tickets with prices $8.75, $6.50, and $6.50, you need to add these values together. Here is the step-by-step process of calculating the total cost:

The expression for the total cost of the three tickets is $21.75.

To find the total cost of the three types of tickets, you sum the individual costs to get a total of $21.75.

To write an expression for the total cost of the three types of tickets, we sum up the cost of each ticket. The three types of tickets cost $8.75, $6.50, and $6.50 each. Therefore, the expression for the total cost is:

Total Cost = $8.75 + $6.50 + $6.50

To calculate the total cost, we add these prices together:

$8.75 + $6.50 = $15.25

$15.25 + $6.50 = $21.75

So, the total cost of the three tickets is $21.75.

Keywords

linear equation, variables

Step 1

Define the variables

Let

x-----> number of erasers

y-----> total cost

we know that

[tex]y=3(0.28)+(0.38)x[/tex]

[tex]y=0.84+(0.38)x[/tex]  ------> this is the linear equation that represent the situation

therefore

the answer is

[tex]y=0.84+(0.38)x[/tex]

Final answer:

The correct answer is option b. percentage, probability, and proportion, because these terms relate to ratios or fractions of a whole and can often be used interchangeably in various statistical contexts. So the correct option is b.

Explanation:

The question is asking which of the following groups contain terms that can be used interchangeably. The correct answer is b. percentage, probability, proportion. These terms can often be used interchangeably because a percentage is a way of expressing a proportion as a number out of 100, and probability is the measure of the likelihood that an event will occur, often expressed as a decimal or a proportion. None of the other options provided (a, c, and d) contain directly interchangeable terms.

Steps:

The expected temperature of the soda after an additional 13 minutes is estimated to be 72°F. This estimation is based on the assumption of a linear temperature increase, calculated from the temperature change observed during the first 5 minutes outside.

The student's question involves the expected temperature of soda after a certain time outside in a hotter environment. Since this scenario does not involve a phase change like melting ice, but rather heating up from the surrounding air, we can infer that Newton's Law of Cooling may be applicable. The Law states that the rate of heat transfer between an object and its surroundings is proportional to the difference in temperature between them. However, without a specific rate of heat transfer given, we cannot apply a formula directly. Therefore, we might assume an approximate linear relationship based on the information provided.

Here's a possible approach:

This logic assumes a simple linear rate, which might not be precisely accurate but should provide an estimated answer.

Final answer:

There is a 40% chance that one of the five doors will have a car behind it as there are two doors with cars and five doors in total.

Explanation:

To find the percentage of doors that have cars behind them in the 21st century version of Let’s Make a Deal with five doors, we can use a simple ratio. Since two out of the five doors have cars behind them, we can set up the ratio as 2 cars to 5 total doors.

The calculation for the percentage would be: (Number of doors with cars / Total number of doors) × 100. Plugging in the numbers gives us:

(2 / 5) × 100 = 40%

Therefore, the percentage of doors with cars behind them is 40%.

The ratio will remain constant. Then the number of cups of sugar used in 1 and 1/2 batches of these cookies will be 1.125.

A ratio is an ordered set of integers a and b expressed as a/b, with b never equaling 0. A percentage is a mathematical expression in which two things are equal.

A recipe for 1 batch of cookies uses 3/4 cup of sugar.

Then the number of cups of sugar used in 1 and 1/2 batches of these cookies will be

We know that the ratio will remain constant. Then we have

1 and 1/2 can be written as 3/2.

Let x be the number of cups of sugar.

Then the ratio will be

[tex]\rm \dfrac{x}{3/2} = \dfrac{3/4}{1}\\\\x = \dfrac{3 \times 3}{2 \times 4}[/tex]

Then we have

x = 9/8

x = 1.125 cups of sugar

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To sketch sin, cos, and tan waves, determine the amplitude and period, plot points on a graph, and connect them smoothly.

The half-life of the element is approximately 542.78 million years.

To find out what percent of the radioactive element remains after 500 million years, we can use the exponential decay formula for radioactive decay:

[tex]\[ N(t) = N_0 \times e^{-kt} \][/tex]

Where:

- [tex]\( N(t) \)[/tex] is the amount of radioactive material remaining at time \( t \)

- [tex]\( N_0 \)[/tex] is the initial amount of radioactive material

- k is the decay constant

- t is the time elapsed

Given that 60% of the radioactive element remains after 400 million years, we know that [tex]\( N(t) = 0.60N_0 \)[/tex]  when [tex]\( t = 400 \)[/tex] million years. We also know that [tex]\( N_0 \)[/tex] represents the initial amount of radioactive material, which will cancel out when we're finding the ratio of remaining material, so we don't need its exact value.

Substituting these values into the exponential decay formula:

[tex]\[ 0.60N_0 = N_0 \times e^{-400k} \][/tex]

We can simplify this to find the decay constant k:

[tex]\[ 0.60 = e^{-400k} \][/tex]

Taking the natural logarithm (ln) of both sides to solve for k:

[tex]\[ \ln(0.60) = -400k \][/tex]

[tex]\[ k = \frac{\ln(0.60)}{-400} \][/tex]

Using this value of k, we can find out what percent of the radioactive element remains after 500 million years:

[tex]\[ N(500) = N_0 \times e^{-500k} \][/tex]

[tex]\[ N(500) = N_0 \times e^{-500 \times \frac{\ln(0.60)}{-400}} \][/tex]

Now, to find the half-life of the element, we know that the half-life [tex](\( T_{\frac{1}{2}} \))[/tex] is the time it takes for the amount of radioactive material to decrease by half. In other words, when [tex]\( N(t) = \frac{1}{2}N_0 \),[/tex] we have:

[tex]\[ \frac{1}{2}N_0 = N_0 \times e^{-kT_{\frac{1}{2}}} \][/tex]

[tex]\[ \frac{1}{2} = e^{-kT_{\frac{1}{2}}} \][/tex]

[tex]\[ \ln\left(\frac{1}{2}\right) = -kT_{\frac{1}{2}} \][/tex]

[tex]\[ T_{\frac{1}{2}} = \frac{\ln(2)}{k} \][/tex]

Now, we can calculate both the percentage remaining after 500 million years and the half-life of the element using the calculated value of k. Let's do that.

First, let's calculate the value of \( k \):

[tex]\[ k = \frac{\ln(0.60)}{-400} \][/tex]

[tex]\[ k ≈ \frac{-0.5108}{-400} \][/tex]

[tex]\[ k = 0.001277 \][/tex]

Now, let's use this value of k to find out what percent of the radioactive element remains after 500 million years:

[tex]\[ N(500) = N_0 \times e^{-500k} \][/tex]

[tex]\[ N(500) = N_0 \times e^{-500 \times 0.001277} \][/tex]

[tex]\[ N(500) ≈ N_0 \times e^{-0.6385} \][/tex]

Now, let's find out what percent this is of the original amount [tex](\( N_0 \))[/tex]:

[tex]\[ \frac{N(500)}{N_0} = e^{-0.6385} \][/tex]

[tex]\[ \frac{N(500)}{N_0} ≈ 0.5274 \][/tex]

So, approximately 52.74% of the radioactive element remains after 500 million years.

Now, let's calculate the half-life of the element:

[tex]\[ T_{\frac{1}{2}} = \frac{\ln(2)}{k} \][/tex]

[tex]\[ T_{\frac{1}{2}} = \frac{\ln(2)}{0.001277} \][/tex]

[tex]\[ T_{\frac{1}{2}} = \frac{0.6931}{0.001277} \][/tex]

[tex]\[ T_{\frac{1}{2}} = 542.78 \text{ million years} \][/tex]

So, the half-life of the element is approximately 542.78 million years.

Answer: The width of the room = 5 meters

The length of the room = 9 meters

Step-by-step explanation:

Let the w denotes the width of the room , then length of the room will be :-

[tex]\text{length}=4+w[/tex]

The formula of perimeter of rectangle :-

[tex]\text{Perimeter}=2(l+w)\\\\\Rightarrow\ 28=2(4+w+w)\\\\\Rightarrow\ 28=2(4+2w)\\\\\Rightarrow\ 28 =8+4w\\\\\Rightarrow\ 4w=20\\\\\Rightarrow\ w=5[/tex]

Hence, The width of the room = 5 meters

The length of the room = 5+4 = 9 meters

The width of the room is 5 meters, and the length is 9 meters.

To find the dimensions of the rectangular room, let's assign variables. Let x be the width of the room. Since the length is 4 meters longer than the width, we can represent the length as x + 4.

Given that the perimeter is 28 meters, we can write an equation: 2(x + 4) + 2x = 28.

Simplifying the equation, we get 4x + 8 = 28. Subtracting 8 from both sides gives 4x = 20. Dividing both sides by 4, we find x = 5. The width of the room is 5 meters, and the length is 5 + 4 = 9 meters.

Answer:

Next two number would be –5, –10, –20,–40, –80, –160.

Step-by-step explanation:

Given : –5, –10, –20, –40, . . .

To find : Use inductive reasoning to describe the pattern. Then find the next two numbers in the pattern.

Solution : We have given that

–5, –10, –20, –40, . . .

We can see from given pattern

-5 * 2 = - 10.

- 10 * 2 = - 20 .

-20 * 2 = - 40

So, each number is multiplied by 2 to get next number.

-40 * 2 = - 80 .

- 80 * 2 = - 160 .

Therefore, Next two number would be –5, –10, –20,–40, –80, –160.

Answer:

[tex]90< \gamma< 180[/tex]

Step-by-step explanation:

An acute angle is an angle that measures more than 0º and less than 90º. And a right angle is an angle that measures exactly 90º. Thus:

Let:

[tex]\alpha = Right\hspace{3}angle=90^{\circ}\\\beta= Acute\hspace{3}angle\hspace{10}0^{\circ}<\beta<90^{\circ}\\\gamma= Sum\hspace{3} of\hspace{3} the\hspace{3} two\hspace{3} angles =\alpha +\beta[/tex]

So, in this sense, the sum of the two angles can´t be equal or greater than 180º, since [tex]\beta<90^{\circ}[/tex] , also it has to be at least greater than 90º since [tex]\beta>0^{\circ}[/tex]

Therefore the sum of the two angles is:

[tex]90^{\circ}< \gamma< 180^{\circ}[/tex]

In another words, the sum is greater than 90º and less than 180º