This is the second practice test adapted for the new Digital SAT format. To help you prepare for the adaptive nature of the exam, the questions are categorized into the four official testing domains: Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry.
Focus: Linear equations, systems of linear equations, and inequalities.
If \(3x - 2 = 41\), what is the value of \(\sqrt{3x + 6}\)?
Instead of solving completely for \(x\), which might result in a fraction, look at the target expression. Notice that we only need the value of \(3x\) to proceed.
First, isolate \(3x\) in the given equation:
\[ 3x = 41 + 2 = 43 \]
Now, substitute \(43\) for the term \(3x\) in the target expression:
\[ \sqrt{3x + 6} = \sqrt{43 + 6} = \sqrt{49} \]
Since the principal square root of 49 is 7, the final answer is 7.
If \(2x + 1 = -x + 3\), what is the value of \(\dfrac{2}{3}x + \dfrac{1}{3}\)?
First, solve the given linear equation for \(x\):
\[ 2x + x = 3 - 1 \]
\[ 3x = 2 \Rightarrow x = \dfrac{2}{3} \]
Now, substitute \(x = \dfrac{2}{3}\) into the target expression:
\[ \dfrac{2}{3}\left(\dfrac{2}{3}\right) + \dfrac{1}{3} = \dfrac{4}{9} + \dfrac{1}{3} \]
To add the fractions, find a common denominator:
\[ \dfrac{4}{9} + \dfrac{3}{9} = \dfrac{7}{9} \]
If \(r\%\) of \(y\) is equal to \(A\), which of the following expressions represents \(y\) in terms of \(r\) and \(A\)?
Translate the word problem into an algebraic equation. Recall that \(r\%\) can be written as the fraction \(\dfrac{r}{100}\):
\[ \left(\dfrac{r}{100}\right) \times y = A \]
To solve for \(y\), multiply both sides of the equation by 100 to eliminate the denominator:
\[ ry = 100A \]
Next, divide both sides by \(r\) to isolate \(y\):
\[ y = \dfrac{100A}{r} \]
Focus: Quadratics, higher-order polynomials, and non-linear equations.
Find the real number \(c\) such that the function \(f(x) = (x - 2)^2 + 4 + c\) has a minimum value equal to 2.
The function is in vertex form, and since the leading coefficient is positive, the parabola opens upward. The minimum value occurs at the vertex.
For any squared real number, the smallest possible value is 0. Thus, the expression \((x - 2)^2\) is at its minimum (0) when \(x = 2\).
At this point, the entire function \(f(x)\) simplifies to its minimum value:
\[ f(2) = 0 + 4 + c = 4 + c \]
Set this minimum value equal to 2, as specified in the problem:
\[ 4 + c = 2 \Rightarrow c = -2 \]
If \(f(x+1) = 2x - 1\), which of the following defines the function \(f(2x)\)?
To find \(f(2x)\), we must first determine the general form of \(f(x)\).
Let the input variable be \(u = x + 1\). This implies that \(x = u - 1\).
Substitute this expression for \(x\) into the original function definition:
\[ f(u) = 2(u - 1) - 1 \]
\[ f(u) = 2u - 2 - 1 = 2u - 3 \]
Therefore, the function rule is \(f(x) = 2x - 3\). Now, evaluate the function for the input \(2x\):
\[ f(2x) = 2(2x) - 3 = 4x - 3 \]
The polynomial \(P(x) = ax^3 + bx^2 + cx + d\) is mathematically equivalent to \((x - 0.2)(x + 1)\left(x + \dfrac{3}{7}\right)\). What is the value of the expression \(-a + b - c + d\)?
This is a conceptual question testing your understanding of polynomial coefficients. Consider what happens if you evaluate the polynomial \(P(x)\) at \(x = -1\):
\[ P(-1) = a(-1)^3 + b(-1)^2 + c(-1) + d = -a + b - c + d \]
The target expression is exactly \(P(-1)\). Since the polynomial is given in a factored form, we can simply evaluate that form at \(x = -1\):
\[ P(-1) = (-1 - 0.2)(-1 + 1)\left(-1 + \dfrac{3}{7}\right) \]
Because the middle factor \((-1 + 1)\) is equal to 0, the entire product becomes 0 regardless of the other terms.
\[ -a + b - c + d = 0 \]
A spherical block of metal weighs 12 pounds. What is the weight, in pounds, of another spherical block of the identical metal if its radius is exactly 3 times the radius of the 12-pound block?
For objects made of the same material, weight is directly proportional to volume. The volume of a sphere is given by \(V = \dfrac{4}{3}\pi r^3\).
When the dimensions of a 3D object are scaled by a factor \(k\), the volume is scaled by \(k^3\). Here, the radius is scaled by \(k = 3\).
\[ \text{New Volume} = 3^3 \times \text{Original Volume} = 27 \times \text{Original Volume} \]
Since the weight scales in the same way:
\[ \text{New Weight} = 12 \times 27 = 324 \text{ pounds} \]
At a high school, each student takes exactly one foreign language. \(\dfrac{3}{8}\) of the students take French, and \(\dfrac{1}{5}\) of the remaining students take German. If all the other students study Italian, what percent of the total student body takes Italian?
Start by determining the portion of students who do not take French:
\[ 1 - \dfrac{3}{8} = \dfrac{5}{8} \]
The portion of the total student body taking German is \(\dfrac{1}{5}\) of this remainder:
\[ \text{German} = \dfrac{1}{5} \times \dfrac{5}{8} = \dfrac{1}{8} \]
Now, subtract both the French and German portions from the whole (1) to find the Italian portion:
\[ \text{Italian} = 1 - \left(\dfrac{3}{8} + \dfrac{1}{8}\right) = 1 - \dfrac{4}{8} = \dfrac{1}{2} \]
Finally, convert the fraction \(\dfrac{1}{2}\) to a percentage: 50%.
A car was purchased for \(\$50,000\). The price decreased by \(\$4,000\) per year for the first two years. After that, it decreased continuously at a constant rate of \(\$6,000\) per year. Which function \(P(t)\) represents the price of the car \(t\) years after the initial 2-year period?
First, find the value of the car at the end of the first 2 years. This will be the "initial" value (y-intercept) for our function \(P(t)\).
\[ \text{Value after 2 years} = 50,000 - 2(4,000) = 50,000 - 8,000 = 42,000 \]
The problem states that after this point, the car loses \(\$6,000\) every year. This constant rate of change represents a linear slope of \(-6,000\).
Using the slope-intercept form (\(y = mx + b\)):
\[ P(t) = 42,000 - 6,000t \]
The lengths of the sides of a triangle are 1.8, 2.4, and 3.0. What is the area of the triangle?
Before using complex formulas, check if the triangle is a right triangle by testing the side ratios. Multiply the sides by 10 to remove decimals: 18, 24, and 30.
Now, divide these by their greatest common factor (6):
\[ 18/6 = 3, \quad 24/6 = 4, \quad 30/6 = 5 \]
Because the sides follow the 3-4-5 Pythagorean triple ratio, the triangle is a right triangle. The two shorter sides (1.8 and 2.4) are the legs, serving as the base and height.
\[ \text{Area} = \dfrac{1}{2} \times \text{base} \times \text{height} = 0.5 \times 1.8 \times 2.4 = 2.16 \]
The graph of the equation \(x^2 + y^2 - 4x + 2y + 2 = 0\) in the xy-plane is a circle. What is the area of the surface enclosed by this circle?
To find the area, we need the radius squared (\(r^2\)). We achieve this by "completing the square" to put the equation in standard form: \((x-h)^2 + (y-k)^2 = r^2\).
Group the \(x\) and \(y\) terms:
\[ (x^2 - 4x) + (y^2 + 2y) = -2 \]
Add the necessary constants to complete each square (half of the middle coefficient, squared):
\[ (x^2 - 4x + 4) + (y^2 + 2y + 1) = -2 + 4 + 1 \]
\[ (x - 2)^2 + (y + 1)^2 = 3 \]
The equation shows that \(r^2 = 3\). The area of a circle is \(\pi r^2\), so the area is \(3\pi\).
Exactly 1 liter of water is poured into a completely empty cylindrical container with a radius of 10 cm. How high, in cm, will the water rise? (\(1 \text{ liter} = 1000 \text{ cm}^3\))
The volume of a cylinder is given by \(V = \pi r^2 h\). We are given the volume (\(1000 \text{ cm}^3\)) and the radius (\(10 \text{ cm}\)).
Substitute these values into the formula:
\[ 1000 = \pi(10)^2 h \]
\[ 1000 = 100\pi h \]
To solve for the height \(h\), divide both sides by \(100\pi\):
\[ h = \dfrac{1000}{100\pi} = \dfrac{10}{\pi} \]