Unit Activity: Advanced Functions

Whenever we have been given to find the product of two square roots, the two roots cancel out each other. This is because [tex] (\sqrt{number} )^2 = number [/tex]

The power of two and the square root cancel out each other

In the given problem we have to multiply the square root 10 by itself, we have

[tex] \sqrt{10} * \sqrt{10} [/tex]

Now, the next step will be

[tex] (\sqrt{10})^2 [/tex]

Now the square root and the power of 2 cancel each other an d the result will be 10

Answer : 10

The solution to the system of equations is [tex]\(x = -3\)[/tex] and [tex]\(y = 2\).[/tex]

let's solve the system of equations using the substitution method.

Given:

1. [tex]\(3x + 2y = -5\)[/tex]

2.[tex]\(2x - 5y = -16\)[/tex]

Step 1: Solve one equation for one variable in terms of the other variable.

Let's solve equation 1 for [tex]\(x\):[/tex]

[tex]\[3x = -2y - 5\][/tex]

[tex]\[x = \frac{-2y - 5}{3}\][/tex]

Step 2: Substitute the expression for [tex]\(x\)[/tex]from step 1 into the other equation.

Substitute [tex]\(x = \frac{-2y - 5}{3}\)[/tex] into equation 2:

[tex]\[2\left(\frac{-2y - 5}{3}\right) - 5y = -16\][/tex]

Step 3: Solve for [tex]\(y\).[/tex]

[tex]\[ \frac{-4y - 10}{3} - 5y = -16 \][/tex]

[tex]\[ -4y - 10 - 15y = -48 \][/tex]

[tex]\[ -19y - 10 = -48 \][/tex]

[tex]\[ -19y = -38 \][/tex]

[tex]\[ y = 2 \][/tex]

Step 4: Substitute the value of [tex]\(y\)[/tex] into one of the original equations to solve for [tex]\(x\).[/tex]

Let's use equation 1:

[tex]\[3x + 2(2) = -5\][/tex]

[tex]\[3x + 4 = -5\][/tex]

[tex]\[3x = -9\][/tex]

[tex]\[x = -3\][/tex]

So, the solution to the system of equations is [tex]\(x = -3\)[/tex] and [tex]\(y = 2\).[/tex]

Complete question:

solve by substitution 3x+2y=-5, 2x-5y=-16

The domain of the function is (-∞, ∞).

Definition of function

A function is an expression that shows the relationship between variables as well as numbers.

The domain of a function is the set of all possible inputs for the function.

From the graph, the domain is the set of all x values, hence:

Domain = (-∞, ∞)

The domain of the function is (-∞, ∞).

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To find the new balance on January 1, we need to calculate the total balance after considering purchases, returns, payments, and finance charges.

Calculate the total charges:

         Total charges = Previous balance + New purchases − Returns

         Total charges = $205.84 + $98.42 − $18.63

         Total charges = $285.63

After the payment, ascertain the extra amount due:

         Unpaid balance = Total charges − Payment

         Unpaid balance = $285.63 − $25

         Unpaid balance = $260.63

Calculate the finance charges:

         Finance charges=Unpaid balance×Monthly interest rate

         Monthly interest rate = 1.5% = 0.015

         Finance charges = $260.63 × 0.015

         Finance charges ≈ $3.91

Calculate the new balance:

         New balance = Unpaid balance + Finance charges

         New balance = $260.63 + $3.91

         New balance = $264.54

So, the new balance on January 1 is approximately $264.54.

To calculate Brent Tyson's new balance on January 1, we first determine the total charges by adding new purchases and subtracting returns from the previous balance.

Then, we deduct his payment to find the unpaid balance.

Next, we calculate finance charges at a rate of 1.5% per month on the unpaid balance.

Finally, we add the finance charges to the unpaid balance to get the new balance.

After these steps, Brent Tyson's new balance on January 1 is $264.54.

This process ensures we account for all transactions and interest charges to accurately determine the updated balance on his Discover card.

Complete Question:

The balance on Brent Tyson’s Discover card on December 1 is $205.84.

In December, he charges an additional $98.42 of purchases, has returns of $18.63, and makes a payment of $25.

If the finance charges are calculated at 1.5% per month on the unpaid balance, find the new balance on January 1.

The tennis player must consecutively land 10 more first serves in bounds to raise their success rate from 62.5% to 70%.

First, let's define the variables:

x = number of additional successful serves needed

40 + x = total number of serves after additional successful serves

25 + x = total number of successful serves after additional successful serves

We set up the equation with the desired success rate of 70%:

(25 + x) / (40 + x) = 0.70

To solve this equation, follow these steps:

Multiply both sides by (40 + x):

25 + x = 0.70 * (40 + x)

Distribute 0.70 on the right side:

25 + x = 28 + 0.70x

Isolate the variable x by subtracting 0.70x from both sides:

25 + x - 0.70x = 28

Combine like terms:

0.30x = 3

Solve for x by dividing both sides by 0.30:

x = 10

Thus, the tennis player must land 10 more consecutive first serves in bounds to increase her success rate to 70%.

Answer:

140 m

Step-by-step explanation:

A kite is a quadrilateral with two distinct pairs of adjacent sides that are equal in length.

The perimeter of the kite is the sum of the length of all its sides. So, the perimeter of kite ABCD can be expressed as:

[tex]\sf Perimeter = AB + BC + CD + DA[/tex]

As BC = AB and CD = DA, then:

[tex]\sf Perimeter = BC + BC + CD + CD\\\\Perimeter = 2(BC + CD)[/tex]

Therefore, to find the perimeter of kite ABCD, we need to determine the lengths of sides BC and CD.

The diagonals of a kite intersect to form two pairs of congruent right triangles, where the sides of the kite are the hypotenuses of these triangles.

in kite ABCD:

The longer diagonal of a kite always bisects the shorter diagonal at a right angle.

Let M be the point of intersection of the diagonals. Therefore:

AM = MC = 24 m

Side BC is the hypotenuse of right triangle BMC, with legs measuring 18 m and 24 m, while side CD is the hypotenuse of right triangle CMD, with legs measuring 32 m and 24 m.

To find the lengths of sides BC and CD, we can use the Pythagorean Theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs.

[tex]\sf BC^2=18^2+24^2 \\\\BC^2=324+576 \\\\BC^2=900\\\\BC=\sqrt{900}\\\\BC=30\; m[/tex]

[tex]\sf CD^2=32^2+24^2 \\\\CD^2=1024+576 \\\\CD^2=1600\\\\CD=\sqrt{1600}\\\\CD=40\; m[/tex]

Now we have determined the length of sides BC and CD, we can substitute them into the perimeter formula:

[tex]\sf Perimeter = 2(30+40) \\\\ Perimeter = 2(70) \\\\ Perimeter = 140 \; m[/tex]

Therefore, the perimeter of kite ABCD is:

[tex]\LARGE\boxed{\boxed{\sf 140\; m}}[/tex]

The solutions of the given quadratic equations is 5/3 and -2

Given the quadratic function expressed as:

Factorizing the function we will have:

6x^2 + 2x – 20 = 0

3x^2 + x - 10 = 0

3x^2 +6x - 5x - 10 = 0

3x(x+2)  5(x+2) = 0

Group the function;

(3x-5)(x+2) = 0

Find the factors

3x- 5 = 0 and x + 2 = 0

x = 5/3 and -2

The solutions of the given quadratic equations is 5/3 and -2

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Since Adam's expenses exceed his earnings, he will have a negative net earnings. Therefore, Adam will not earn any money this season. In fact, he will have a net loss of $7.50.

To calculate how much Adam will earn this season,  subtract his expenses from his total earnings.

Earnings per mowing session: $20

Number of mowing sessions in the grass-growing season: 18 weeks / 2 weeks = 9 sessions

Total earnings: $20 per session * 9 sessions = $180

Expenses:

Cost of the lawn mower: $165

Cost of gasoline per session: $2.50 per session * 9 sessions = $22.50

Total expenses: $165 + $22.50 = $187.50

To find Adam's net earnings, subtract his expenses from his total earnings:

Net earnings = Total earnings - Total expenses

Net earnings = $180 - $187.50

Net earnings = -$7.50

Since Adam's expenses exceed his earnings, he will have a negative net earnings. Therefore, Adam will not earn any money this season. In fact, he will have a net loss of $7.50.

Adam charges $20 to mow his neighbors lawn every 2 weeks for the 18 weeks of grass-growing season. He wants to buy a nice lawn mower for $165 and gasoline costs $2.50 every 2 weeks.  How much will Adam earn this season?

For 52/9 to be expressed as a decimal, you need to divide the fraction.

52/9 = 5.78 (rounded by the 1hundreds in the decimals place).

The graph of g(x) will be the same as the graph of f(x) shifted vertically downwards by 5 units because k=-5 represents a vertical shift, not a horizontal one. There is no horizontal shift involved.

When considering the functions [tex]f(x)=2^x[/tex]and g(x)=f(x)+k, with k=-5, we look at how the graph of g(x) compares to the graph of f(x).

Since the value of k is negative, it implies that a constant is being subtracted from the function f(x). This means that the graph of g(x) will not be shifted horizontally, but rather vertically downwards by 5 units.

Therefore, option c is the correct answer: the graph is not shifted horizontally from the graph of f(x).

The greatest possible error when measuring to the nearest quarter of an inch is:

1/8 in.

Error in measurement is the difference between the measured value of a quantity and its true value. It is an inevitable part of any measurement, and there are many different sources of error.

The greatest possible error when measuring to the nearest quarter of an inch is 1/8 of an inch.

This is because the smallest measurement interval is 1/4 of an inch, and the greatest possible error is half of the smallest measurement interval.

Therefore, the greatest possible error when measuring to the nearest quarter of an inch is 1/8 in.

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